GC 303 
.S7 
Copy 1 




OCEAN TIDES 



WITH 

ELABORATE TABLES 

SHOWING 

FLUCTUATIONS OF THE SURFACE OF 
THE OCEAN AT ALL POINTS 



BY 
JOHN NELSON STOCKWELL. Ph. D. 



'M 



Digitized by the Internet Archive 
in 2011 with funding from 
The Library of Congress 



http://www.arGhive.org/details/oceantidesnewsolOOstoc 



OCEAN TIDES 



A NEW SOLUTION OF THE PROBLEM OF THE TIDES, WITH ELABORATE 
TABLES, SHOWING BY MERE INSPECTION, THE NATURE AND AMOUNT 
OF THE FLUCTUATIONS OF THE SURFACE OF THE OCEAN AT 
ALL PLACES PRODUCED BY THE ATTRACTIVE FORCES OF 
THE SUN AND MOON; TOGETHER WITH THE VARIATION • 
OF GRAVITY AT ALL PLACES ON LAND, PRO- 
DUCED BY THE SAME FORCES. 



BY 

JOHN NELSON STOCKWELL, M. A., Ph. D. 



Multum in parvo 



1919 
Press of THOS. P. NICHOLS & SON CO. 

LYNN, MASSACHUSETTS 






Copyright 1919 
By JOHN NELSON STOCKWELL, M. A.. Ph. D. 

CLEVELAND, OHIO 



©CI:a529813 



-V'.: I 



PREFACE 

IT IS NOT the purpose of this book to give a new theory of the Tides. 
The cause of the Tides was sufficiently and correctly explained by 
Sir Isaac Newton in the year 1687; and the mathematical develop- 
ment of the effects produced by that cause upon the waters of the 
Ocean has been the great unsolved problem before the scientific world 
for more than two hundred and thirty years. The most celebrated and 
illustrious mathematicians of the world have long devoted their great 
talents to its solution; and the prince of mathematicians and theoret- 
ical astronomers, Laplace, pronounces the "problem of the Tides" 
the most difficult problem of celestial mechanics. 

But notwithstanding the admitted difficulty of the problem, I 
have here given two simple though rigorous solutions of the problem, 
made by means of entirely independent methods; and it is believed 
that a more complete knowledge in regard to the operation of the 
tidal forces will also contribute valuable information for the improve- 
ment of the sciences of meteorology and terrestrial physics. 

John N. Stockwell. 
Cleveland, April 18 1919. 



y. 



INTRODUCTORY AND HISTORICAL 

THE PROBLEM of the Tides has probably been the occasion of 
more laborious and troublesome research among mathematicians 
than any other problem of astronomy. Although the cause of the 
Tides has been known from the earliest ages, no philosopher or mathe- 
matician has yet been able to construct a system of theoretical Tide 
tables that have any practical value. Pliny speaks of the Tides with 
as much familiarity and confidence as does Sir Isaac Newton and 
other modern philosophers, and attributes them to the influence of the 
Sun and Moon, without, however, explaining the nature of that influ- 
ence nor by what means it is communicated to the waters of the ocean. 
The discovery of the principle of universal gravitation by Newton, 
however, invested the problem with new interest, by showing the 
nature of the physical relation which exists between the Earth and 
Moon, and thus brought the problem within the domain of strict 
mathematical calculation. According to the law of gravitation the 
attractive force of one body upon every particle of another body varies 
inversely as the square of the distance between the centers of the two 
bodies; and acts in the line directed towards itself. But in the appli- 
cation of this law to physical problems it is nearly always necessary to 
find the force of attraction in all directions; and it is chiefly owing to 
the complications arising from this necessity, that makes the problem 
so difficult. I shall now attend to the resolution of the force in the 
required directions, and will give two solutions of the problem; — a 
geometrical solution and an algebraic solution. 

I first give a geometrical resolution of the disturbing force; and, 
in order to assist the imagination, I have constructed a diagram to 
illustrate the operation of the disturbing force. 





-^ 


f 


\\ ^^^"'^^^^'--'^Jt^ 




\ 


^""^--^^ /V 




/ 




\^^--?4\^ 


/ ^ 


/^\ 




\s /a'\ ^^~^--^^ 



6 OCEAN TIDES 

In the diagram, A denotes the center of the Earth, m, the center 
of the Moon; and the distance between A and m is denoted by a\ 
F, denotes any fixed point on the Earth's surface, at which we may 
suppose a movable particle or marble to be placed. As the Moon 
attracts the center of the Earth through the distance AG = h, the 
fixed point P on the surface will describe the line PK, which is equal 
and parallel to AG; and the movable particle at P will describe the 
line PJ ; and will be distant from the point P by the length of the 
line KJ . If we then draw through K the line HK equal and parallel 
to PJ , and through H, the line HP equal and parallel to KJ , we shall 
have the parallelogram which corresponds to the proposed problem; 
and it is evident that the whole disturbance will be measured by the 
length and direction of the line HP = A', which we will now determine. 

If we denote the Moon's distance from the attracted point by A, 
and its distance from the center of the Earth by a\ the force at the 

point P in the direction Pm will be measured by -^ = a; and the 

force acting in the direction PK will be measured by -^ = b; in 
which the mass of the Moon is supposed to be equal to unity. 

Now the diagram shows, — (the radius of the Earth being unity) 

A2 = a'2 + I - 2a' cos (3, (i) tan tt = , ^^^ ^ ^ , (2) 

a — cos 

A' sin ^' = rt sin TT } . . „„ ^ ^/ a sin tt , . 

o> K C ' (3) Whence tan ^' - (4) 

A cos ^ = ^ — a cos TT ) b — a cos tt 

, ^ , a sin T b — a cos x . s 

and A' = . ^, = -, — (5) 

sm j8 cos /3 

The amount and direction of the disturbing force is therefore fully 
determined for any point of the surface of the earth. 

Now equations (3) give A'^ = a^ + ^^ — 2ab cos tt , (6) 

a' — cos (3 

1 
I -\- a'^ — 2a' cos jSp 



-r» • / N • <^ ~" cos p ,. . 

Jiut equation (2) gives cos tt = : ; (7; 



and equation (6) becomes finally 



2\a' — cos /35 



A'2 ^ a" J^¥ -^ -^^^^-^ . (8) 



a'Mi + ^'' - 2a' cos ^j' 
This equation gives the whole disturbance in terms of known quantities. 



OCEAN TIDES 7 

I will now show that equation (8) is correct by means of the hori- 
zontal and vertical components of the force acting at the point P. 

According to writers on celestial mechanics the horizontal and 
vertical components of a force acting upon a point as P, in the diagram 
are given by the equations, 

80 = + --^ , (9) 

\i -\- a'' - 2a' cos 0\'^' 

I — a' cos , cos 3 , , 

hr = ^ ^ + — ^ ; do) 

\\ +a'2 - 2a' cos ^p 
and the whole disturbance would be given by the expression 

Now equations (9) and (10) give 
^^^ a''^ sin^ i8 sin^ |8 2a' sin^ , . 



Ji + a'2 — 2a' COS/55 u. /o w I /2 ^/ /D? 2 

(Z ^ )i + a ^ — 2a coSjSi 

2 _ I ~ 2a' cos /3 + a'^ cos^ /3 cos^ \2 — 2a' cos jSj cos , . 

§1 +a'2- 2a' cos /3p + ~^^ + ,., , ,„ ^ ^ 
^ '^ a'2Ji + a'2 — 2a'cosi8S^ 

The sum of the first terms of these equations is equal to a^; the sum 
of the second terms is equal to lr\ and the sum of the last terms is 
equal to the last term of equation (8); and the two solutions give 
identically the same result, and are therefore correct. 

We shall now give a numerical example of the use of the formulas 
we have already computed. The radius of the Earth being unity or i, 
we shall suppose the Moon^s distance a' = 60. Equations (i) and 
(2) will then give the values of A and tt corresponding to any assumed 
values of /3. The forces a and b will be given by the equations 

a = ^2 ^^d ^ ^ " ^ ' 

the Moon's mass being supposed equal to unity. We can then form 
equations (3) from which we get A' and /3' as in the following table. 



OCEAN TIDES 



Table I 



Showing amount and direction of forces at the Earth's surface produced by a 
body at the Moons distance. 



3 


Whole force 

A' 


Direction 


Components of force 


Vertical 


Horizontal 


o° o' o" 


5I94960 


180° 0' 


+ 5194960 


— 5I00000 


5 o 


5)94676 


177 26 


5)93867 


5)12470 


lO o 


5)93826 


174 51 


5)90558 


5)24546 


20 


5190490 


169 26 


5)77910 


5)46024 


30 


5)85155 


163 30 


5)58622 


5)61763 


40 


5)78184 


156 42 


5)35134 


5)69846 


44 24 20 


5)74722 


153 18 


5)24198 


5 1 70695 


45 


5)74242 


152 49 


+ 5)22718 


5)70681 


54 20 36 


5)66431 


144 21 


5]ooooo 


5)66431 


60 


5)61675 


138 18 


-5)12603 


5)60574 


80 


5)48362 


108 5 


5)42667 


-5)22768 


89 31 21 


5)46296 


89 31 


5)46296 


5)00000 


90 


5)46291 


88 34 


5)46290 


+ 61 1156 


100 


5)48316 


69 15 


5)41522 


5)24705 


no 


5)53642 


52 51 


-5)29094 


5)45066 


124 52 24 


5)64537 


34 53 


5]ooooo 


5)64537 


130 


5)68437 


30 7 


+ 5)11761 


5)67442 


134 24 44 


5)71743 


26 26 


5)22150 


5)68229 


135 


5)72173 


25 58 


5)23536 


5)68224 


150 


5)81822 


15 43 


5)57122 


5)58581 


160 


5)86465 


10 4 


5)74830 


5)43322 


170 


5)89351 


4 55 


5)86346 


5)22998 


175 


5)90084 


2 27 


5)89329 


+ 5I11670 


180 


5)90328 





+ 5)90328 


5)00000 



Figures in brackets indicate the number of ciphers between the decimal point and first 
significant figure of number, thus 5)81822 = 0.0000081822. 



The horizontal and vertical components of the force are 
then given by equations (9) and (10) as in the last two columns 
of the table; and we shall find that the numbers in the last two columns 
will give the values of A' in the second column. But .the most impor- 
tant and surprising fact made known by this computation, and shown 
by the table, relates to the direction in which the disturbing force acts; 
which is directly contrary to that claimed by Newton, and which has 
been universally accepted as true, by the scientific world for more 
than two hundred and thirty years. In other words, Newton claimed 
that the disturbing force is always directed towards the line passing 
through the centers of the Earth and the disturbing body; whereas 
the actual computation as given in the table, shows that it acts in 
the opposite direction. It is therefore confidently believed that this 
fundamental error committed by Newton, and persistently repeated 
by his successors from generation to generation, has been the cause 



OCEANTIDES 9 

of the long stationary condition, or even retrograde advance of tidal 
science; and it is also believed that a correction of the errors incidentally 
growing out of that fundamental error, will have a beneficial effect on 
the future development of tidal science; and the correction of the 
fundamental error committed by Newton is the primary purpose of 
this work. 

A closer examination of Table I shows that for. the arguments 
i8 = o°o'o", 89° 31^21" and 180° o'o'', the horizontal component 
of the disturbing force vanishes; but in these cases the whole dis- 
turbing force becomes vertical, as shown in the table. It also shows 
that for the arguments ^ = 54° 20' 36" and /3 = 124° 52' 24", the 
vertical force vanishes, and the whole disturbing force becomes hor- 
izontal, as shown in the table; and what is still more important, the 
table shows that the horizontal force always tends to move the water 
towards a point of greater gravity; and when the point of greatest gravity 
is reached, the horizontal force vanishes. All these conditions shov\^ 
the perfect harmony existing in the operation of the forces of Nature, 
and further exemplifies the harmony of truth. The perfect agreement 
of these two methods of computation affords a sufficient guarantee of 
their correctness. 

I will now explain the construction of Table II, which shows the 
character and amount of the lunar tidal wave. I have given in 
Table I the increment of gravity produced by a unit mass acting at a 
distance of sixty semi-diameters of the Earth; and this is called the 
vertical component of the disturbing force in Table I. But the values 
are given in terms of the semi-diameter of the Earth as the unit of 
linear measure; and as the mind cannot easily grasp the significance 
of numbers referred to such a base I have multiplied them by the 
Earth^s radius expressed in feet, which is asumed to be 20887680. 
And since the mass of the Earth is supposed to be 81 times that of 
the Moon; we must divide the numbers in the table by 81 in order 
to get the force produced by the Moon. If we therefore multiply 
the numbers in the column of vertical components by —128936.3 we 
shall have the height of the lunar tidal wave expressed in feet, as 
in Table II for the distance a' = 60.0, and the corresponding values 
of the argument j3. 

Since the eccentricity of the Moon's orbit is 0.0549, the Moon's 
actual distance will always be included within the distances 56.5 and 
63.5 semi-diameters of the Earth; and I have therefore computed 
Table II for distances differing by one-half a semi-diameter from 
each other between those limits. The tides for distances not given 
in the Table may therefore be easily found by interpolation from the 
given tabular numbers. But since the Moon is so near the Earth, 
the tides on the opposite sides of the Earth are quite different in magni- 
tude, and it has been found necessary to compute them for values of 
the argument /3 between 0° and 180°.- 



10 



OCEAN TIDES 



Table II 
LUNAR TIDAL WAVE 



-9 


a' 56.5 


a' 57.0 


a' 57.5 


a' 58.0 


a' 58.5 


a' 59.0 


a' 59.5 


a' 60.0 


/5 


o° 


-1.4686 


-1.4300 


-1.3926 


-1.3566 


-1. 3219 


-1.2882 


-1.2558 


-1.2244 


0° 


I 


1.4680 


1-4293 


1-3919 


1-3559 


1. 3213 


1.2876 


1.2552 


1.2238 


I 


2 


1.4659 


1.4274 


1.3900 


1-3541 


1.3194 


1.2858 


1.2534 


1.2220 


2 


3 


1.4626 


1.4247 


1-3857 


1-3509 


1. 3163 


1.2828 


1.2505 


1.2191 


3 


4 


1-4577 


1-4193 


1.3822 


1-3465 


1. 3120 


1.2786 


1.2464 


1.2151 


4 


5 


-1-4513 


-1-4133 


-1-3764 


— 1.3408 


— 1.3064 


-1.2732 


— 1. 2410 


-1.2099 


5 


6 


1-4437 


1.4059 


1.3692 


1.3338 


1.2996 


1.2666 


1.2346 


1.2037 


6 


7 


1-4347 


1-3973 


1.3609 


1-3257 


1. 2916 


1.2588 


1.2270 


1. 1963 


7 


8 


1.4246 


1-3879 


I-35I2 


1. 3162 


1.2824 


1.2498 


1. 2183 


1. 1879 


8 


9 


1-4131 


1.3766 


1.3402 


1-3056 


1.2724 


1-2395 


1.2084 


1. 1784 


9 


lO 


-1.4004 


-1. 3641 


-1.3279 


-1.2936 


— 1.2604 


-1.2282 


-1-1975 


-1. 1678 


10 


II 


1-3863 


1-3504 


1-3146 


1.2807 


1.2478 


1.2161 


1-1854 


1. 1560 


II 


12 


1-3709 


1-3348 


1.3000 


1.2664 


1-2339 


1.2030 


1. 1724 


1-1430 


12 


13 


1-3541 


1. 3 1 86 


1.2842 


1.2510 


1. 2189 


1. 1883 


1.1581 


1.1291 


13 


14 


1.3362 


1. 3012 


1.2672 


1.2344 


1.2028 


1. 1728 


1. 1428 


1.1141 


14 


15 


-1-3171 


-1.2827 


-1.2492 


— 1. 2169 


-1-1857 


-1-1557 


-1. 1265 


— 1.0982 


15 


i6 


1.2969 


1.2630 


1.2300 


1. 1982 


1. 1675 


1. 1378 


1. 1092 


1.0815 


16 


17 


1-2753 


1. 2421 


1.2096 


1. 1784 


1. 1482 


1.1192 


1.0908 


1.0636 


17 


i8 


1.2528 


1. 2201 


1.1881 


1. 1576 


1. 1278 


1.0994 


1. 0718 


1.0447 


18 


19 


1. 2291 


T.1962 


1. 1658 


1-1357 


1. 1066 


3.0788 


1. 0514 


1.0250 


19 


20 


-1.2045 


-1. 1729 


-1. 1423 


-1.1128 


— 1.0844 


-1.0570 


-1-0303 


-1.0045 


20 


21 


1. 1786 


1. 1469 


1.1178 


1. 0881 


1.0611 


1-0343 


1.0082 


0.9829 


21 


22 


1.1518 


1. 1202 


1. 0924 


1.0629 


1.0370 


1. 0108 


0.9852 


0.9606 


22 


23 


1.1241 


1.0929 


1.0662 


1.0370 


1.0121 


0.9865 


0.9616 


0.9376 


23 


24 


1-0955 


1.0650 


1. 0391 


1. 0105 


0.9863 


0.9614 


0.9372 


0.9138 


24 


25 


-1.0658 


— 1.0364 


— 1. 0109 


-0.9837 


-0.9596 


-0.9352 


— 0.9119 


-0.8892 


25 


26 


1-0354 


1.0075 


0.9820 


0.9561 


0.9321 


0.9085 


0.8858 


0.8637 


26 


27 


1.0042 


0.9773 


0.9524 


0.9277 


0.9040 


0.8811 


0.8592 


0.8377 


27 


28 


0.9723 


0.9467 


0.9221 


0.8984 


0.8753 


0.8532 


0.8318 


0.8111 


28 


29 


0.9394 


0.9144 


0.8909 


0.8678 


0.8457 


0.8242 


0.8036 


0.7838 


29 


30 


-0.9058 


-0.8820 


-0.8592 


-0.8370 


-0.8155 


-0.7951 


-0.7751 


— 0.7558 


30 


31 


0.8718 


0.8486 


0.8269 


0.8054 


0.7849 


0.7649 


0.7459 


0.7274 


31 


32 


0.8371 


0.8152 


0.7940 


0.7736 


0.7538 


0.7347 


0.7164 


0.6985 


32 


33 


0.8015 


0.7805 


0.7604 


0.7407 


0.7216 


0.7035 


0.6860 


0.6690 


33 


34 


0.7657 


0.7456 


0.7265 


0.7070 


0.6895 


0.6722 


0.6553 


0.6389 


34 


35 


-0.7293 


— 0.7102 


— 0.6922 


— 0.6740 


— 0.6566 


— 0.6403 


-0.6242 


-0.6088 


35 


36 


0.6928 


0.6747 


0.6574 


0.6403 


0.6240 


0.6083 


0.5930 


0.5782 


36' 


37 


0-6555 


0.6387 


0.6219 


0.6053 


0.5902 


0.5755 


0.5609 


0.5471 


37 


38 


0.6180 


0.6020 


0.5864 


0.5709 


0.5568 


0.5427 


0.5 2 88 


0.5159 


38 


39 


0.5803 


0.5655 


0.5505 


0.5358 


0.5226 


0.5095 


0.4967 


0.4849 


39 



OCEAN TIDES 



11 



Table II {Continued) 
LUNAR TIDAL WAVE 



^ 


fl'56.5 


c'57.0 


«'57.5 


a' 58.0 


a' S^.S 


a' 59.0 


a' 59.5 


a> 60.0 


P 


40° 


-0.5425 


— 0.5282 


-0.5147 


— 0.5014 


-0.4889 


-0.4764 


— 0.4646 


-0.4530 


40- 


41 


0.5041 


0.4910 


0.4783 


0.4660 


0.4543 


0.4427 


0.4318 


0.4210 


41 


42 


0.4657 


0.4536 


0.4419 


0.4305 


0.4197 


0.4090 


0.3990 


0.3890 


42 


43 


0.4273 


0.4163 


0.4055 


0.3951 


0.3851 


0-3753 


0.3662 


0.3570 


43 


44 


0.3889 


0.3788 


0.3690 


0.3596 


0.3506 


0.3417 


o-iUi 


0.3250 


44 


45 


-0.3505 


— 0.3416 


-0.3325 


-0.3241 


-0.3158 


— 0.3077 


-0.3005 


-0.2930 


45 


46 


0.3121 


0.3044 


0.2958 


0.2887 


0.2810 


0.2741 


0.2677 


0.2607 


46 


47 


0.2737 


0.2672 


0.2598 


0.2532 


0.2466 


0.2444 


0.2349 


0.2289 


47 


48 


0.2353 


0.2297 


0.2234 


0.2178 


0.2122 


0.2070 


0.2020 


0.1971 


48 


49 


0.1973 


0.1923 


0.1872 


0.1830 


0.1778 


0.1740 


0.1698 


0.1654 


49 


50 


— 0.1598 


-0.1553 


-0.1513 


— 0.1482 


-0.1437 


— 0.1408 


-0.1376 


-0.1337 


50 


51 


0.1225 


0.1186 


0.1158 


0.1134 


O.IIOO 


0.1078 


0.1054 


0.1025 


51 


52 


0.0848 


0.0825 


0.0806 


0.0787 


0.0768 


0.0749 


0.0731 


0.0714 


52 


SZ 


0.0481 


0.0469 


0.0457 


0.0447 


0.0435 


0.0429 


0.0416 


0.0408 


53 


54 


— 0.0114 


— 0.0114 


— O.OIIO 


— 0.0109 


— 0.0107 


— 0.0108 


— 0.0104 


—0.0102 


54 


55 


+0.0245 


+0.0236 


+0.0223 


+0.0221 


+0.0216 


+0.0205 


+0.0202 


+0.0195 


55 


56 


0.0598 


0.0581 


0.0565 


0.0549 


0.0534 


0.0519 


0.0505 


0.0492 


56 


57 


0.0949 


0.0912 


0.0898 


0.0871 


0.0850 


0.0825 


0.0802 


0.0782 


57 


58 


0.1291 


0.1242 


0.1228 


0.1189 


O.I 1 59 


0.1127 


0.1097 


0.1070 


58 


59 


0.1627 


0.1573 


O.I54I 


0.1498 


0.1462 


0.1421 


0.1384 


0-1351 


59 


60 


+0.1956 


+0.1904 


+0.1853 


+0.1804 


+0.1757 


+0.1711 


+0.1668 


+0.1625 


60 


61 


0.2279 


0.2217 


0.2160 


0.2102 


0.2049 


0.1993 


0.1945 


0.1895 


61 


62 


0.2594 


0.2524 


0.2457 


0.2392 


0.2332 


0.2271 


0.2215 


0.2159 


62 


63 


0.2899 


0.2823 


0.2748 


0.2677 


0.2609 


0.2541 


0.2477 


0.2414 


63 


64 


0.3198 


0.3116 


0.3031 


0.2953 


0.2876 


0.2807 


0.2732 


0.2663 


64 


65 


+0.3489 


+0.3397 


+0.3307 


+0.3223 


+0.3139 


+0.3065 


+0.2981 


+0.2906 


65 


66 


0.3770 


0.3670 


0.3573 


0.3481 


0.3392 


0.3318 


0.3221 


0.3140 


66 


67 


0.4041 


0.3932 


0.3831 


0.3732 


0.3637 


0.3564 


0.3451 


0.3368 


67 


68 


0.4303 


0.4189 


0.4079 


0.3974 


0.3872 


0.3804 


0.3677 


0.3585 


68 


69 


0.4555 


0.4435 


0.4320 


0.4202 


0.4101 


0.4016 


0.3894 


0.3786 


69 


70 


+0.4797 


+0.4672 


+0.4551 


+0.4428 


+0.4318 


+0.4220 


+0.4101 


+0.4000 


70 


71 


0.5027 


0.4896 


0.4770 


0.4639 


0.4527 


0.4415 


0.4300 


0.4191 


71 


72 


0.5248 


0.5110 


0.4979 


0.4848 


0.4724 


0.4602 


0.4488 


0.4376 


72 


73 


0.5460 


0.5317 


0.5179 


0.5048 


0.4914 


0.4777 


0.4668 


0.4551 


73 


74 


0.5657 


0.5508 


0.5370 


0.5240 


0.5091 


0.4946 


0.4837 


0.4717 


74 ' 


75 


+0.5845 


+0.5692 


+0.5543 


+0.5413 


+ 0.5260 


+0.5103 


+ 0.4998 


+0.4873 


75 


76 


0.6018 


0.5859 


0.5706 


0.5578 


0.5416 


0.5254 


0.5146 


0.5018 


76 


77 


0.6182 


0.6017 


0.5861 


0.5723 


0.5564 


0.5400 


0.5287 


0.5155 


77 


78 


0.6331 


0.6164 


0.6004 


0.5856 


0.5700 


0.5537 


0.5416 


0.5281 


78 


79 


0.6470 


0.6298 


0.6135 


0.5981 


0.5825 


0.5667 


0.5535 


0.5396 


79 



12 



OCEAN TIDES 



Table II {Continued) 
LUNAR TIDAL WAVE 



;y 


1 

a' 56.5 


a' 51.{) 


a' 57.5 


a' 58.0 


«'58.5 


a' 59.0 


a' 59.5 

1 


a' 60.0 


^ 


80° 


+ 0.6594 


+ 0.6421 


+ 0.6254 


+ 0.6093 


+ 0.5938 


+0.5787 


1 
+0.5642 


+0.5501 


So° 


81 


0.6709 


0.6532 


0.6361 


0.6199 


0.6041 


0.5888 


0.5740 


0.5597 


81 


82 


0.6808 


0.6629 


0.6457 


0.6294 


0.6131 


0.5977 


0.5826 


0.5681 


82 


83 


0.6897 


0.6715 


0.6540 


0.6375 


0.6213 


0.6054 


0.5903 


0.5754 


83 


84 


0.6970 


0.6788 1 


0.6612 


0.6444 


0.6278 


0.6118 


0.5965 


C.5818 


84 


85 


+ 0.7033 


+0.6847 


+ 0.6670 


+ 0.6508 


+ 0.6336 


+ 0.6186 


— 0.6019 


+0.5868 


85 


86 


0.7081 


0.6893 


0.6716 


0.6556 


0.6378 


0.6236 


0.6061 


0.590?; 


86 


87 


0.7118 


0.6929 


0.6751 


0.6582 


0.6412 


0.6262 


0.6093 


0.5940 


87 


88 


0.7139 


0.6953 


0.6773 


0.6599 


0.6432 


0.6269 


0.61 12 


0.5961 


88 


89 


+ 0.7130 


+0.6962 


+ 0.6782 


+ 0.6608 


+0.6441 


+ 0.6278 


+ 0.6121 


+0.5969 


89 


90 


+ 0.7133 


+0.6959 


+ 0.6775 


+ 0.6605 


+0.6437 


+ 0.6275 


+0.6123 


+0.5967 


90 


91 


0.7129 


0.6943 


0.6763 


0.6590 


0.6422 


0.6262 


0.6109 


0.5954 


91 


92 


0.7099 


0.6914 


0.6736 


0.6564 


0.6396 


0.6236 


0.6080 


0.5929 


92 


93 


0.7058 


0.6872 


0.6696 


0.6524 


0.6358 


0.6200 


0.6041 


0.5893 


93 


94 


0.7002 


0.6818 


0.6644 


0.6473 


0.6309 


0.6150 


0.5993 


0.5846 


94 


95 


0.6939 


0.6752 


0.658c 


0.6410 


0.6248 


0.6092 


0.5937 


0.5791 


95 


96 


0.6853 


0.6675 


0.6503 


0.6336 


0.6175 


0.6021 


0.5871 


0.5725 


96 


97 


0.6757 


0.6584 


0.6433 


0.6246 


0.6090 


0.5939 


0.5791 


0.5647 


97 


98 


0.6650 


0.6479 


0.6313 


0.6145 


0.5994 


0.5845 


0.5700 


0.5559 


98 


99 


0.6531 


0.6366 


0.6201 


0.6040 


0.5888 


0.5743 


0.5599 


0.5463 


99 


100 


0.6402 


0.6240 


0.6079 


0.5923 


0.5770 


0.5629 


0.5489 


0.5354 


100 


lOI 


0.6261 


0.6102 


0.5945 


0.5795 


0.5644 


0-5505 


0.5368 


0.5234 


lOI 


102 


0.6108 


0.5952 


0.5799 


0.5655 


0.5505 


0.5369 


0.5236 


0.5107 


102 


103 


0.5940 


0.5792 


0.5643 


- 0.5503 


0.5359 


0.5225 


0.5096 


0.4969 


103 


104 


0.5768 


0.5619 


0.5475 


0.5338 


0.5200 


0.5070 


0.4944 


0.4822 


104 


105 


0.5579 


0.5438 


0.5293 


0.5163 


0.5031 


0.4905 


0.4782 


0.4665 


105 


106 


0.5380 


0.5242 


0.5100 


0.4980 


0.4851 


0.4730 


0.4613 


0.4499 


106 


107 


0.5171 


0.5041 


0.4901 


0.4785 


0.4659 


0.4547 


0.4434 


0.4329 


107 


108 


0.4952 


0.4825 


0.4693 


0.4582 


0.4466 


0.4355 


0.4246 


0.4142 


108 


109 


0.4722 


0.4599 


0.4478 


0.4368 


0.4258 


0.4158 


0.4049 


0.3950 


109 


no 


0.4482 


0.4367 


0.4252 


0.4147 


0.4042 


0.3943 


0.3844 


0.3750 


no 


III 


0.4234 


0.4124 


0.4020 


0.3917 


0.3818 


0.3725 


0.3631 


0.3543 


III 


112 


0.3977 


0.3875 


0.3777 


0.3680 


0.3587 


0.3499 


0.3412 


0.3328 


112 


113 


0.3709 


0.3607 


0.3521 


0.3430 


0.3346 


0.3261 


0.3181 


0.3104 


113 


114 


0.3434 


0.3344 


0.3259 


0.3171 


0.3097 


0.3021 


0.2944 


0.2875 


114 


115 


0.3150 


0.3063 


0.2989 


0.2914 


0.2843 


0.2770 


0.2701 


0.2638 


115 


116 


0.2860 


0.2787 


0.2714 


0.2647 


0.2581 


0.2517 


0.2451 


0.2396 


116 


117 


0.2560 


0.2493 


0.2430 


0.2370 


0.2311 


0.2252 


0.2195 


0.2146 


117 


118 


0.2254 


0.2194 


0.2140 


0.2089 


0.2035 


0.1985 


0.1934 


0.1890 


118 


119 


0.1942 


0.1890 


0.1844 


0.1800 


O.I7S4 


0.1710 


0.1671 


0.1630 


119 



OCEAN TIDES 



13 



Table II {Continued) 
LUNAR TIDAL WAVE 



p 


c'56.5 


fl'57.0 


c'57.5 


a' 58.0 


a' S%.S 


a' 59.0 

1 


a' 59.5 


a' 60.0 


[^ 


1 20° 


0.1624 


0.1582 


0.1543 


0.1506 


0.1467 


0.1432 


0.1397 


0.1364 


120° 


121 


0.1298 


0.1267 


0.1234 


0.1204 


0.1172 


0.1 143 


0.1117 


0.1092 


121 


122 


0.0967 


0.0949 


0.0920 


0.0898 


0.0875 


0.0855 


0.0834 


0.0816 


122 


123 


0.0633 


0.0623 


0.0603 


0.0588 


0.0572 


0.0558 


0.0547 


0.0536 


123 


124 


+0.0294 


+0.0294 


+0.0281 


+ 0.0275 


+0.0269 


+0.0263 


+0.0257 


+0.0252 


124 


125 


— 0.0051 


— 0.0046 


— 0.0047 


— 0.0045 


— 0.0042 


— 0.0044 


-0.0039 


-0.0038 


125 


126 


0.0398 


0.0387 


0.0376 


0.0367 


0.0355 


0.0347 


0.0338 


0.0325 


126 


127 


0.0749 


0.0732 


0.0710 


0.0692 


0.0676 


0.0657 


0.0639 


0.0617 


127 


128 


0.1103 


0.1077 


0.1045 


0.1018 


0.0992 


0.0966 


0.0941 


0.0913 


128 


129 


0.1464 


0.1431 


0.1387 


0-1351 


0.1317 


0.1282 


0.1250 


0.1213 


129 


130 


0.1824 


0.1745 


0.1729 


0.1684 


0.1642 


0.1600 


O.T559 


0.1516 


130 


131 


0.2187 


0.2140 


0.2071 


0.2017 


0.1967 


0.1916 


0.1868 


0.1818 


131 


132 


0.2547 


0.2480 


0.2414 


0.2351 


0.2292 


0.2233 


0.2176 


0.2121 


132 


133 


0.2911 


0.2834 


0.2756 


0.2688 


0.2621 


0.2553 


0.2485 


0.2425 


133 


134 


-0.3274 


— 0.3188 


— 0.3098 


— 0.3025 


— 0.2950 


-0.2871 


-0.2794 


-0.2731 


134 


135 


-0.3636 


-0.3542 


-0.3449 


-0.3358 


-0.3276 


-0.3193 


— 0.3104 


-0.3035 


135 


136 


0.4001 


0.3897 


0.3795 


0.3698 


0.3604 


0.3513 


0.3423 


0.3339 


136 


137 


0.4364 


0.4249 


0.4143 


0.4028 


0.3932 


0.3832 


0.3734 


0.3644 


^2,7 


138 


0.4726 


0.4604 


0.4489 


0.4359 


0.4259 


0.4154 


0.4045 


0.3951 


138 


139 


0.5086 


0.4951 


0.4829 


0.4694 


0.4581 


0.4465 


0.4352 


0.4248 


139 


140 


-0.5442 


-0.5301 


— 0.5166 


-0.5031 


-0.4904 


-0.4780 


— 0.4660 


-0.4545 


140 


141 


0-5799 


0.5649 


0.5502 


0.5356 


0.5225 


0.5090 


0.4965 


0.4842 


141 


142 


0.6153 


0.5992 


0.5837 


0.5680 


0.5544 


0.5400 


0.5269 


0.5139 


142 


143 


0.6501 


0.6332 


0.6166 


0.6005 


0.5857 


0.5706 


0.5566 


0.5429 


143 


144 • 


0.6845 


0.6666 


0.6493 


0.6328 


0.6168 


0.6012 


0.5862 


0.5718 


144 


145 


-0.7186 


— 0.6998 


-0.6818 


— 0.6692 


— 0.6476 


— 0.6310 


— 0.6152 


— 0.6002 


145 


146 


0.7523 


0.7327 


0.7138 


0.6956 


0.6779 


0.6610 


0.6438 


0.6285 


146 


147 


0.7852 


0.7648 


0.7452 


0.7260 


0.7078 


0.6897 


0.6720 


0.6560 


147 


148 


0.8178 


0.7966 


0.7761 


0.7563 


0.7372 


0.7186 


0.6999 


0.6834 


148 


149 


0.8500 


0.8279 


0.8068 


0.7863 


0.7662 


0.7466 


0.7278 


0.7160 


149 


150 


-0.8811 


-0.8586 


-0.8365 


-0.8154 


-0.7953 


-0.7745 


-0.7550 


— 0.7366 


150 


151 


0.9122 


0.8885 


0.8658 


0.8438 


0.8232 


0.8013 


0.7816 


0.7622 


151 


152 


0.9421 


0.9177 


0.8941 


0.8712 


0.8503 


0.8279 


0.8074 


0.7874 


152 


153 


0.9716 


0.9464 


0.9221 


0.8978 


0.8765 


0.8529 


0.8325 


0.8120 


153 


154 


1.0002 


0.9742 


0.9491 


0.9230 


0.9021 


0.8785 


0.8571 


0.8360 


154 


15s 


— 1.0280 


— 1.0030 


-0.9755 


— 0.9482 


— 0.9270 


— 0.9025 


-0.8811 


— 0.8642 


155 


156 


1.0549 


T.0275 


1. 0010 


0.9729 


0.9510 


0.9271 


0.9040 


0.8818 


156 


157 


1.0811 


1.0530 


1. 0261 


0.9984 


0.9747 


0.9517 


0.9266 


0.9038 


157 


158 


1. 1065 


1.0777 


1. 0501 


1.0224 


0.9974 


0.9752 


0.9483 


0.9249 


158 


159 


1-1307 


1. 1 003 


1.0733 


I.0457 


1.0195 


0.9973 


0.9692 


0.9452 


159 



14 



OCEAN TIDES 



Table II {Continued) 
LUNAR TIDAL WAVE 



/5 


fl'56.5 


a' 51.0 


w'57.5 


a' 58.0 


«'58.5 


a' 59.0 


«'59.5 


a' 60.0 


/? 


1 60° 


-1.1541 


-1.1241 


-I-0953 


— 1.0674 


— 1.0405 


— 1. 0182 


— 0.9892 


— 0.9648 


160° 


161 


1. 1766 


1.1461 


1.1168 


1.0884 


1.0608 


1.0369 


1.0085 


0.9836 


t6i 


162 


1. 1979 


1.1671 


1.1371 


1. 1082 


T.0800 


1.0548 


1.0270 


1. 0017 


162 


163 


1. 2184 


1.1871 


1-1564 


1. 1272 


1.0986 


1. 0719 


1.0447 


1. 0187 


163 


164 


1.^378 


1.2058 


1. 1748 


1. 1449 


I.I 160 


1.0882 


1.0611 


1-0350 


164 


165 


-1.2562 


— 1.2236 


— 1. 1922 


— 1. 1620 


-1. 1327 


-1. 1044 


-1.0768 


-1-0503 


165 


166 


1.2736 


1.2405 


1.2086 


1-1778 


1. 1482 


1.1193 


1. 0916 


1.0649 


166 


167 


1.2896 


1. 2561 


1.2239 


1.1931 


1. 1629 


1-1335 


1. 1054 


1. 0781 


167 


168 


1.3046 


1.2707 


1.2382 


1.2067 


1. 1764 


I. 1465 


1.1184 


1.0909 


168 


169 


1. 3186 


1.2842 


1-2515 


1. 2196 


1. 1889 


I. 1589 


1. 1303 


1. 1025 


169 


170 


-1-3314 


-1.2967 


— 1.2636 


-1. 2317 


— 1.2004 


-1. 1705 


-1.1414 


-1-1133 


170 


171 


1-3430 


1.3080 


1.2747 


1.2423 


1. 2 109 


1. 1806 


1-1513 


1. 1229 


171 


172 


1-3534 


1. 3182 


1.2844 


1. 2521 


1.2202 


1. 1898 


1. 1602 


1.1317 


172 


173 


1.3624 


1.3272 


1. 2931 


1.2603 


1.2284 


1. 1980 


1.1681 


1-1393 


173 


174 


1-3704 


1-3348 


1.3007 


1.2676 


1-2357 


1.2050 


1. 1748 


1. 1460 


174 


175 


-1.3770 


-1. 3416 


-1-3071 


-1.2738 


— 1. 2418 


— 1.2114 


— 1. 1807 


-1-1517 


175 


176 


1.3827 


1-3469 


1-3125 


1. 2791 


1.2468 


1. 2157 


1-1855 


1. 1564 


176 


177 


1.3869 


1-3523 


1. 3166 


1-2833 


1.2508 


1. 2199 


1. 1892 


1. 1600 


177 


178 


1.3900 


I-354I 


1-3195 


1.2865 


1-2535 


1.2220 


1.1918 


1. 1626 


178 


179 


1-3919 


1.3560 


1-3213 


1.2888 


1-2552 


1.2238 


1-1936 


1.1641 


179 


180 


-1.3926 


-1-3566 


-1-3219 


— 1.2900 


-1-2558 


-1.2244 


— 1. 1 940 


— 1. 1646 


180 



OCEAN TIDES 



15 



Table II {Continued) 
LUNAR TIDAL WAVE 



^ 


a' 60.0 


a' 60.5 


a'6\.0 


a'61.5 


«'62.0 


a' 62.5 


«'63.0 


a' 63.5 


ft 


o° 


-1.2244 


-I.I943 


— 1. 1646 


-1. 1363 


— 1. 1088 


— 1. 0821 


-1.0564 


-1-0314 


0" 


I 


1.2238 


1-1936 


1. 1642 


1. 1357 


1. 1082 


1. 08 1 6 


1.0560 


1.0309 


I 


2 


1.2220 


1.1919 


1. 1627 


1.1341 


1. 1067 


1.0791 


1.0545 


1.0295 


2 


3 


1.2191 


1. 1890 


I. 1600 


1.1314 


1. 1 04 1 


1.0775 


1. 0521 


1. 0271 


3 


4 


1.2151 


1.1851 


r.1563 


1. 1277 


1. 1005 


1.0740 


1.0485 


1.0238 


4 


5 


— 1.2099 


— 1. 1800 


— 1.1511 


— I.I 230 


— 1.0960 


— 1.0695 


-1. 0441 


— 1. 0194 


5 


6 


1.2037 


1.1741 


1. 1449 


1.1170 


1. 0901 


1.0640 


1.0386 


1. 0140 


6 


7 


1.1963 


1. 1667 


1-1377 


1.1103 


1. 0816 


1.0575 


1.0323 


1.0078 


7 


8 


1. 1879 


1.15S6 


1. 1299 


1. 1024 


1.0758 


1.0500 


1.0249 


1.0007 


8 


9 


1. 1784 


1.1491 


1. 1207 


1-0935 


1.0668 


1.0415 


1. 0167 


0.9926 


9 


lO 


-1. 1678 


-1. 1386 


-1.1105 


-1.0835 


-I.0573 


-1.0320 


-1.0073 


-0.9836 


10 


II 


1. 1560 


1.1271 


1.0993 


1.0727 


1.0465 


1. 0217 


0.9974 


0.9737 


II 


12 


1-1430 


1.1147 


1.0872 


1.0608 


1.0352 


1. 0103 


0.9862 


0.9630 


12 


13 


1.1291 


I.IOII 


1.0738 


1.0477 


1.0229 


0.9980 


0.9742 


0.9506 


13 


14 


1.1141 


1.0866 


1.0599 


1.0338 


1.0096 


0.9848 


0.9613 


0.9375 


14 


15 


— 1.0982 


— 1.0711 


— 1.0446 


— 1.0191 


-0.9954 


— 0.9708 


-0.9477 


-0.9239 


15 


i6 


1. 0815 


1.0547 


1.0288 


1.0037 


0.9802 


0.9560 


0.9332 


0.9096 


16 


17 


1.0636 


1.0373 


1.0117 


0.9870 


0.9641 


0.9402 


0.9177 


0.8950 


17 


18 


1.0447 


1. 0189 


0.9938 


0.9694 


0.9469 


0.9235 


0.9015 


0.8795 


18 


19 


1.0250 


0.9999 


0.9751 


0.9512 


0.9290 


0.9060 


0.8846 


0.8633 


19 


20 


-1.0045 


-0.9797 


-0.9555 


-0.9323 


— 0.9100 


-0.8880 


-0.8669 


-0.8465 


20 


21 


0.9829 


0.9588 


0.9352 


0.9124 


0.8906 


0.8688 


0.8484 


0.8282 


21 


22 


0.9606 


0-9370 


0.9139 


0.8917 


0.8704 


0.8489 


0.8294 


0.8095 


22 


23 


0.9376 


0.9146 


0.8922 


0.8703 


0.8494 


0.8285 


0.8100 


0.7900 


23 


24 


0.9138 


0.8912 


0.8694 


0.8482 


0.8276 


0.8074 


0.7891 


0.7701 


24 


25 


-0.8892 


-0.8672 


-0.8458 


-0.8252 


— 0.8054 


— 0.7860 


— 0.7676 


-0.7493 


25 


26 


0.8637 


0.8423 


0.8218 


0.8017 


0.7825 


0.7637 


0.7456 


0.7279 


26 


27 


0.8377 


0.8170 


0.7970 


0.7776 


0.7590 


0.7408 


0.7231 


0.7061 


27 


28 


0.8111 


0.7910 


0.7717 


0.7529 


0.7348 


0.7171 


0.7001 


0.6837 


38 


29 


0.7838 


0.7643 


0.7455 


0.7274 


0.7096 


0.6928 


0.6765 


0.6606 


29 


30 


-0.7558 


-0.7370 


-0.7184 


-0.7016 


— 0.6842 


-0.6682 


-0.6524 


-0.6371 


30 


31 


0.7274 


0.7094 


0.6919 


0.6752 


0-6583 


0.643T 


0.6280 


0.6132 


31 


32 


0.6985 


0.6812 


0.6645 


0.6485 


0.6322 


0.6177 


0.6030 


0.5889 


32 


33 


0.6690 


0.6523 


0.6364 


0.6210 


0.6054 


0.5916 


0.5775 


0.5641 


33 


34 


' 0.6389 


0.6232 


0.6082 


0-5933 


0.5786 


0.5653 


0.5519 


0.5391 


34 


35 


-0.6088 


-0.5936 


-0.5793 


-0.5652 


-0.5513 


-0.5386 


-0.525'^ 


-0.5138 


3S 


36 


0.5782 


0.5640 


0.5503 


0-5369 


0.5240 


0.5116 


0.499s 


0.4879 


36 


37 


0.5471 


0.5336 


0.5205 


0.5081 


0.4956 


0.4840 


0.4727 


0.4615 


37 


38 


0.5159 


0.5032 


0.4910 


0.4791 


0.4675 


0.4565 


0.4458 


0.4350 


3^ 


39 


0.4849 


0.4726 


0.4609 


0.4S00 


0.4389 


0.4286 


0.4187 


0.4086 


39 



16 



OCEAN TIDES 



Table II {Continued) 
LUNAR TIDAL WAVE 



3 


a' 60.0 


a' 60.5 


«'6L0 


fl'6L5 


fl'62.0 


c'62.5 


«'63.0 


a' 63.5 


/? 


400 


-0-4530 


— 0.4418 


-0.4310 


-0.4207 


— 0.4106 


— 0.4008 


-0.3914 


-0.3821 


40° 


41 


0.4210 


0.4107 


0.4006 


0.3910 


0.3816 


0.3725 


0.3638 


0.3552 


41 


42 


0.3890 


0.3797 


0.3702 


0.3613 


0.3526 


0.3442 


0.3362 


0.3283 


42 


43 


0.3570 


0.3485 


0.3398 


0.3316 


0.3236 


0.3159 


0.3086 


0.3014 


43 


44 


0.3250 


0.3175 


0.3094 


0.3019 


0.2947 


0.2877 


0.2809 


0.2744 


44 


45 


-0.2930 


-0.2860 


-0.2790 


— 0.2720 


-0.2657 


-0.2504 


-0.2533 


— 0.2474 


45 


46 


0.2607 


0.2546 


0.2485 


0.2422 


0.2367 


0.2311 


0.2259 


0.2204 


46 


47 


0.2289 


0.2234 


0.2180 


0.2126 


0.2077 


0.2028 


0.1981 


0.1934 


47 


48 


0.1 07 1 


0.1922 


0.1876 


0.1830 


0.1788 


0.1745 


0.1704 


0.1666 


48 


49 


0.1654 


0.1612 


0.1576 


0.1535 


0.150] 


0.1468 


0.1431 


0.1403 


49 


50 


-0.1337 


-0.1303 


— 0.1276 


— 0.1242 


— 0.1215 


— 0.1191 


— 0.1158 


— 0.1140 


50 


51 


0.1025 


0.0998 


0.0976 


0.0953 


0,0933 


0.0914 


0.0890 


0.0873 


51 


52 


0.0714 


0.0696 


0.0681 


0.0666 


0.0651 


0.0635 


0.0621 


0.0607 


52 


53 


0.0408 


0.0396 


0.0386 


0.0380 


0.0372 


. 0.0364 


0.0354 


0.0357 


53 


54 


— 0.0102 


— 0.0099 


— 0.0096 


— 0.0099 


— 0.0093 


— 0.0094 


— 0.0090 


— 0.0091 


54 


55 


+0.0195 


+ 0.0192 


+ 0.0192 


+ 0.0178 


+0.0175 


+ 0.0169 


+0.0166 


+0.0167 


55 


56 


0.0492 


0.0479 


0.0475 


0.0452 


0.0442 


0.0431 


0.0420 


0.0425 


56 


57 


0.0782 


0.0764 


0.0750 


0.0724 


0.0705 


0.0687 


0.0662 


0.0653 


57 


58 


0.1070 


0.1043 


0.1020 


0.0991 


0.0965 


0.0941 


0.0900 


0.0885 


58 


59 


0-1351 


0.1316 


0.1285 


0.1251 


0.1218 


0.1187 


0.1141 


0.1123 


59 


60 


+ 0.1625 


+ 0.1583 


+ 0.1545 


+ 0.1506 


+0.1468 


+ 0.1432 


+0.1378 


+ 0.1364 


60 


61 


0.1895 


0.1848 


0.1802 


0.1753 


0.1713 


0.1672 


0.1610 


0.1589 


61 


62 


0.2159 


0.2103 


0.2053 


0.1996 


0.1992 


0.1905 


0.1840 


0.1810 


62 


63 


0.2414 


0.2352 


0.2295 


0.2231 


0.2184 


0.2132 


0.2069 


0.2027 


63 


64 


0.2663 


0.2596 


0.2532 


0.2460 


0.2410 


0.2351 


0.2295 


0.2241 


64 


65 


+ 0.2906 


+0.2834 


+ 0.2764 


+0.2685 


+0.2631 


+0.2566 


+0.2501 


+0.2446 


65 


66 


0.3140 


0.3065 


0.2987 


0.2909 


0.2844 


0.2773 


0.2706 


0.2649 


66 


67 


0.3368 


0.3286 


0.3203 


0.3123 


0.3049 


0.2974 


0.2900 


0.2836 


67 


68 


0-3585 


0.3499 


0.3410 


0.3327 


0.3246 


0.3167 


0.3091 


0.3020 


68 


69 


0.3786 


0.3703 


0.3610 


0.3521 


0.3437 


0.3355 


0.3274 


0.3196 


69 


70 


+ 0.4000 


+ 0.3899 


+ 0.3804 


+ 0.3708 


+0.3620 


+0.3533 


+0.3449 


+0.3367 


70 


71 


0.4191 


0.4087 


0.3985 


0.3887 


0.3795 


0.3705 


0.3616 


0.3529 


71 


72 


0.4376 


0.4267 


0.4161 


0.4060 


0.3961 


0.3867 


0.3775 


0.3685 


72 


73 


0-4551 


0.4442 


0.4325 


0.4226 


0.4119 


0.4023 


0.3926 


0.3885 


73 


74 


0.4717 


0.4605 


0.4482 


0.4380 


0.4270 


0.4173 


0.4070 


0.3974' 


74 


75 


+0.4873 


+ 0.4760 


+ 0.4630 


+ 0.4526 


+0.4411 


+ 0.4307 


+0.4204 


+ 0.4107 


75 


76 


0.5018 


0.4904 


0.4770 


0.4658 


0.4545 


0.4436 


0.4330 


0.4229 


76 


77 


0-5155 


0.5035 


0.4903 


0.4786 


0.4670 


0.4557 


0.4454 


0.4345 


77 


78 


0.5281 


0.5155 


0.5023 


0.4902 


0.4784 


0.4663 


0.4568 


0.4451 


.78 


79 


0-5396 


0.5266 


0.5135 


0.5010 


0.4890 


0.4771 


0.4664 


0.4549 


79 



OCEAN TIDES 



17 



Table II {Continued) 
LUNAR TIDAL WAVE 



^ 


a' 60.0 


a' 60.5 


fl'6L0 


1 
a'dl.S 


a' 62.0 


a 62.5 

1 


1 

a' 63.0 


fl'63.5 


;9 


80° 


+0.5501 


-1-0.5366 


+ 0.5235 


+0.5105 


+ 0.4984 


+ 0.4865 


+ 0.4749 


+ 0.4638 


80° 


81 


0.5597 


0.5459 


0.5325 


0.5195 


0.5072 


0.4948 


0.4832 


0.4716 


81 


82 


0.5681 


0.5541 


0.5406 


0.5273 


0.5147 


0.5Q22 


0.4905 


0.4784 


82 


83 


0.5754 


0.5613 


0.5475 


0.5343 


0.5216 


0.5088 


0.4970 


0.4847 


83 


84 


0.5818 


0.5674 


0.5535 


0.5401 


0.5271 


0.5145 


0.5024 


0.4899 


84 


85 


+0.5868 


+0.5725 


+ 0.5582 


+0.5446 


+ 0.5319 


+ 0.5193 


+0.5070 


+ 0.4947 


85 


86 


0.5908 


0.5766 


0.5619 


0.5485 


0.5357 


0.5231 


0.5105 


0.4983 


86 


87 


0.5940 


0.5795 


0.5651 


0.5513 


0.5391 


0.5259 


0.5132 


0.5009 


87 


88 


0.5961 


0.5814 


0.5672 


0.5535 


0.5402 


0.5277 


0.5149 


0.5028 


88 


89 


+0.5969 


+0.5822 


+ 0.5680 


+ 0.5540 


+ 0.5409 


+ 0.5281 


+ 0.5156 


+ 0.5036 


89 


90 


+0.5967 


+0.5819 


+ 0.5678 


+ 0.5541 


+ 0.5407 


+ 0.5279 


+ 0.5154 


+ 0.5034 


90 


91 


0.5954 


0.5806 


0.5666 


0.5529 


0.5396 


0.5270 


0.5143 


0.5020 


91 


92 


0.5929 


0.5783 


0.5644 


0.5507 


0.5375 


0.5255 


0.5123 


0.4996 


92 


93 


0.5893 


0.5749 


0.5610 


0.5476 


0.5348 


0.5221 


0.5089 


0.4969 


93 


94 


0.5846 


0.5705 


0.5566 


0.5433 


0.5302 


0.5178 


0.5048 


0.4931 


94 


95 


+0.5791 


+0.5650 


+ 0.5512 


+0.5382 


+ 0.5251 


+ 0.5127 


+ 0.5001 


+0.4889 


95 


96 


0.5725 


0.5585 


0.5449 


0.5318 


0.5191 


0.5068 


0.4948 


0.4832 


96 


97 


0.5647 


0.5510 


0.5375 


0.5247 


0.5119 


0.5000 


0.4880 


0.4768 


97 


98 


0.5559 


0.5424 


0.5291 


0.5164 


0.5040 


0.4922 


0.4795 


0.4693 


98 


99 


0.5463 


0.5329 


0.5199 


0.5075 


0.4957 


0.4836 


0.4720 


0.4612 


99 


100 


+0.5354 


+0.5222 


+0.5096 


+ 0.4974 


+ 0.4854 


+ 0.4740 


+0.462.8 


+ 0.4520 


100 


lOI 


0.5234 


0.5109 


0.4982 


0.4868 


0.4744 


0.4636 


0.4525 


0.4420 


lOI 


102 


0.5107 


0.4983 


0.4864 


0.4749 


0.4625 


0.4522 


0.4415 


0.4310 


102 


103 


0.4969 


0.4850 


0.4731 


0.4624 


0.4499 


0.4400 


0.4296 


0.4196 


103 


i 104 


0.4822 


0.4704 


0.4594 


0.4486 


0.4364 


0.4270 


0.4170 


0.4072 


104 


105 


+0.4665 


+0.4544 


+ 0.4437 


+0.4342 


+0.4220 


+ 0.4128 


+ 0.4035 


+ 0.3941 


105 


106 


0.4499 


0.4370 


0.4285 


0.4186 


0.4071 


0.3979 


0.3891 


0.3801 


106 


107 


0.4329 


0.4197 


0.4112 


0.4022 


0.3917 


0.3824 


0.3741 


0.3657 


107 


108 


0.4142 


0.4014 


0.3944 


0.3846 


0.3758 


0.3662 


0.3583 


0.3500 


108 


109 


0.3950 


0.3834 


0.3753 


0.3670 


0.3580 


0.3498 


0.3417 


0.3340 


109 


no 


+0.3750 


+0.3643 


+ 0.3568 


+0.3485 


+ 0.3399 


+ 0.3320 


+0.3245 


+0.3171 


no 


III 


0.3543 


0.3454 


0.3370 


0.3296 


0.3211 


0.3138 


0.3066 


0.2996 


III 


112 


0.3328 


0.3248 


0.3169 


0.3093 


0.3021 


0.2949 


0.2880 


0.2812 


112 


113 


0.3104 


0.3031 


0.2954 


0.2885 


0.2816 


0.2757 


0.2688 


0.2624 


113 


114 


0.2875 


0.2805 


■ 0.2735 


0.2671 


0.2607 


0.2549 


0.2489 


0.2432 


114 


IIS 


+0.2638 


+0.2576 


+0.2510 


+0.2451 


+ 0.2398 


+ 0.2339 


+0.2285 


+0.2232 


115 


116 


0.2396 


0.2338 


0.2282 


0.2224 


0.2174 


0.2125 


0.2074 


0.2026 


116 


117 


0.2146 


0.2092 


0.2042 


0.1992 


0.1946 


0.1902 


0.1857 


0.1815 


117 


118 


0.1890 


0.1844 


0.1800 


0.1756 


0.1716 


0.1676 


0.1637 


0.1601 


118 


119 


0.1630 


0.1588 


0.1551 


0.1514 


0.1479 


0.1444 


0.1411 


0.1381 


119 



18 



OCEAN TIDES 



Table II (Continued) 
LUNAR TIDAL WAVE 



/5 


a' 60.0 


a' 60.5 


a' 61.0 


a' 61.5 


a' 62.0 


a' 62.5 


a' 63.0 


a' 63.5 


/? 


120° 


+0.1364 


+0.1331 


+0.1300 


+0.1269 


+0.1240 


+0.1210 


+0.1183 


+0.1157 


120° 


121 


0.1092 


0.1065 


0.1040 


0.1016 


0.0995 


0.0969 


0.0948 


0.0927 


121 


122 


0.0816 


0.0797 


0.0778 


0.0760 


0.0747 


0.0726 


0.0710 


0.0695 


122 


123 


0.0536 


0.0523 


0.0511 


0.0500 


0.0491 


0.0478 


0.0468 


0.0457 


123 


124 


+0.0252 


-0.0247 


+0.0242 


+0.0237 


+0.0232 


+0.0228 


+0.0222 


+0.0218 


124 


125 


— 0.0038 


— 0.0036 


-0.0037 


— 0.0034 


-0.0028 


— 0.0026 


— 0.0027 


— 0.0025 


125 


126 


0.0325 


0.0319 


0.0310 


0.0301 


0.0290 


0.0286 


0.0278 


0.0270 


126 


127 


0.0617 


0.0606 


0.0593 


0.0575 


0.0558 


0.0550 


0.0533 


0.0519 


127 


128 


0.0913 


0.0894 


0.0871 


0.0849 


0.0828 


0.0809 


0.0789 


0.0770 


128 


129 


0.1213 


0.1188 


0.1158 


0.1129 


0.1096 


0.1075 


0.1049 


0.1024 


129 


130 


— 0.1516 


— 0.1482 


— 0.1441 


-0.1409 


-0.1368 


-0.1342 


— 0.1309 


— 0.1278 


130 


131 


0.1818 


0.1776 


0.1731 


0.1689 


0.1643 


0.1609 


0.1570 


0.1532 


131 


132 


0.2121 


0.2069 


0.2017 


0.1969 


0.1921 


0.1875 


0.1831 


0.1787 


132 


133 


0.2425 


0.2365 


0.2307 


0.2252 


0.2197 


0.2144 


0.2094 


0.2044 


133 


134 


0.2731 


0.2661 


0.2597 


0.2535 


0.2471 


0.2414 


0.2357 


0.2301 


134 


135 


-0.3035 


-0.2957 


-0.2887 


— 0.2818 


-0.2749 


-0.2684 


— 0.2620 


-0.2558 


135 


136 


0.3339 


0.3252 


0.3177 


0.3102 


0.3025 


0.2953 


0.2884 


0.2815 


136 


137 


0.3644 


0.3550 


0.3467 


0.3384 


0.3299 


0.3222 


0.3144 


0.3070 


T37 


13S 


0.3951 


0.3847 


0.3755 


0.3666 


0.3576 


0.3491 


0.3407 


0.3^25 


138 


139 


0.4248 


0.4150 


0.4041 


0.3948 


0.3846 


0.3756 


0.3667 


0.3580 


139 


140 


-0.4545 


-0.4433 


-0.4325 


-0.4231 


— 0.4118 


— 0.4021 


-0.3926 


-0.3835 


140 


141 


0.4842 


0.4725 


0.4608 


0.4504 


_ 0.4389 


0.4283 


0.4186 


0.4086 


141 


142 


0.5139 


0.5015 


0.4889 


0.4776 


' 0.4657 


0.4544 


0.4445 


0.4336 


142 


143 


0.5429 


0.5297 


0.5166 


0.5044 


0.4922 


0.4804 


0.4697 


0.4581 


143 


144 


0.5718 


0.5576 


0.5441 


0.5310 


0.5183 


0.5060 


0.4949 


0.4825 


144 


145 


— 0.6002 


-0.5852 


-0.5713 


-0.5574 


-0.5442 


-0.5314 


— 0.5196 


— 0.5066 


145 


146 


0.6285 


0.6128 


0.5982 


0.5836 


0.5698 


0.5562 


0.5438 


0.5304 


146 


147 


0.6560 


0.6398 


0.6245 


0.6095 


0.5949 


0.5808 


0.5675 


0.5538 


147 


148 


0.6834 


0.6666 


0.6505 


0.6347 


0.6196 


0.6049 


0.5907 


0.5769 


148 


149 


0.7140 


0.6927 


0.6762 


0.6598 


0.6441 


0.6286 


0.6141 


0.5996 


149 


150 


— 0.7366 


-0.7185 


— 0.7012 


— 0.6843 


— 0.6679 


-0.6518 


— 0.6369 


— 0.6219 


150 


151 


0.7622 


0.7436 


0.7267 


0.7082 


0.6912 


0.6748 


0.6591 


0.6434 


151 


152 


0.7874 


0.7682 


0.7495 


0.7315 


0.7139 


0.6974 


0.6807 


0.6648 


152 


153 


0.8120 


0.7919 


0.7730 


0-7539 


0.7363 


0.7190 


0.7020 


0.6856 


153 


154 


0.8360 


0.8154 


0.7957 


0.7759 


0.7581 


0.7401 


0.7227 


0.7058 


154 


155 


-0.8642 


-0.8379 


— 0.8179 


-0.7973 


-0.7792 


— 0.7608 


-0.7428 


-0.7253 


155 


156 


0.8818 


0.8602 


0.8393 


0.8182 


0.7997 


0.7807 


0.7623 


0.7446 


156 


157 


0.9038 


0.8824 


0.8603 


0.8388 


0.8197 


0.8003 


0.7815 


0.7631 


157 


158 


0.9249 


0.9035 


0.8805 


0.8588 


0.8388 


0.8187 


0.7997 


0.7810 


158 


1^0 


0.9452 


0.9238 


0.89Q9 


0.8780 


0.8572 


0.8369 


0.8175 


0.7982 


159 



OCEAN TIDES 



19 



Table II (Continued) 
LUNAR TIDAL WAVE 



,5 


a' 60.0 


a' 60.5 


a' 61.0 


fl'61.5 


fl'62.0 


fl'62.5 


«'63.0 


a' 63.5 


.3 


1 60° 


— 0.9648 


-0.9428 


— 0.9185 


-0.8965 


-0.8747 


-0.8543 


-0.8343 


-0.8148 


160° 


161 


0.9836 


0.9609 


0-9365 


0.9138 


0.8919 


0.8711 


0.8506 


0.8308 


161 


162 


1. 0017 


0.9778 


0.9536 


0-9305 


0.9083 


0.8868 


0.8662 


0.8460 


162 


163 


1. 0187 


0.9943 


0.9699 


0.9466 


0.9229 


0.9021 


0.8809 


0.8606 


163 


164 


1-0350 


1.0097 


0.9852 


0.9616 


0.9386 


0.9164 


0.8950 


0.8742 


164 


165 


-1-0503 


— 1.0250 


-0.9999 


-0-9759 


-0.9522 


— 0.9301 


— 0.9083 


-0.8874 


165 


166 


1.0649 


1.0390 


1-0137 


0.9893 


0-9654 


0.9428 


0.9209 


0.8997 


166 


167 


1. 0781 


1-0523 


1.0265 


1. 0019 


0.9778 


0-9550 


0-9325 


0.9112 


167 


168 


1.0909 


1.0643 


1.0386 


1.0136 


0.9843 


0.9662 


0.9434 


0.9218 


168 


169 


1. 1025 


I-0753 


1.0497 


1.0244 


1. 0000 


0.9767 


0-9535 


0.9315 


169 


170 


-1-1133 


-1-0853 


— 1. 0601 


-1.0342 


— 1.0099 


-0.9859 


-0.9628 


-0.9403 


170 


171 


1. 1229 


1.0947 


1.0692 


1-0434 


1. 0183 


0.9946 


0.9711 


0.9486 


171 


172 


1.1317 


1. 1030 


1.0776 


1-0515 


1.0264 


1.0020 


0.9787 


0-9559 


172 


173 


1-1393 


1.1109 


1.0849 


1.0588 


1-0332 


1.0090 


0-9855 


0.9623 


173 


174 


1. 1460 


1.1177 


1. 0912 


1.0649 


I-0393 


1. 0147 


0.9913 


0.9679 


174 


17s 


-1-1517 


-1-1235 


-1.0965 


— 1.0702 


-1.0444 


— 1.0200 


-0.9962 


-0.9727 


175 


176 


1. 1564 


1.1281 


1. 1008 


1.0742 


1.0488 


1.0238 


1. 0001 


0.9768 


176 


177 


1. 1 600 


1-1315 


1. 1043 


1.0778 


1-0519 


1.0273 


1-0033 


0.9798 


177 


178 


1. 1626 


1-1338 


1. 1068 


1.0802 


1-0543 


1.0294 


1.0054 


0.9819 


178 


179 


1.1641 


1-1355 


1.1081 


1. 0817 


1-0557 


1. 0310 


1.0067 


0.9834 


179 


180 


— 1. 1646 


— 1. 1360 


-1. 1088 


-1. 0821 


-1.0564 


-1. 0314 


— 1.0072 


-0.9839 


180 



20 



OCEAN TIDES 



In the construction of Table III for the Sun's tidal wave we have 
supposed the Sun's mean distance (a'), to be 23,500 times the semi- 
diameter of the Earth; and the Sun's mass m' = 328,000, the Earth's 
mass being unity. The Sun's distance is so great that the tides on 
the opposite sides of the Earth are very nearly equal; and as the eccen- 
tricity of the Earth's orbit is only 0.0167712, it follows that the Sun's 
actual distance at any time will always be included between the limits 
23100 and 23900 times the semi-diameter of the Earth. I have there- 
fore computed the Sun's tidal wave for seven different distances, 
differing by one hundred semi-diameters of the Earth from each other, 
between those limits. Table III will therefore afford the means of 
getting the Sun's tidal wave for all possible distances of the Sun from 
the Earth. 

The change of gravity at the Earth's surface arising from the 
Tidal forces may be found by dividing the tabular height of the tide, 
expressed in feet, by one-half of the Earth's radius expressed in feet, 
and changing the algebraic sign of the quotient; or, change of gravity 
= — 8r ^ 10443840, the undisturbed gravity at the surface being 
denoted bv — i. 



Table III 
SOLAR TIDAL WAVE 



;^ 


a' 23200 


a' 23300 


a' 23400 


a' 23500 


a' 23600 


a' 23700 


a' 23800 


^ 


0° 


— 0.5486 


-0.5416 


-0.5347 


-0.5279 


-0.5212 


— 0.5146 


— 0.5082 


180* 


I 


0.5483 


0.5413 


0.5344 


0.5277 


0.5209 


0.5143 


0.5079 


179 


2 


0.5476 


0.5406 


0.5337 


0.5270 


0.5202 


0.5137 


0.5073 


178 


3 


0.5463 


0.5393 


0.5324 


0.5258 


0.5190 


0.5125 


0.5064 


177 


4 


0.5446 


0.5377 


0.5308 


0.5241 


0.5174 


0.5109 


0.5045 


176 


5 


-0.5423 


-0.5353 


-0.5286 


-0.5215 


-0.5153 


-0.5087 


-0.5024 


175 


6 


0.5396 


0.5327 


0.5258 


0.5193 


0.5127 


0.5062 


0.4999 


174 


7 


0.5363 


0.5294 


0.5228 


0.5162 


0.5097 


0.5031 


0.4969 


173 


8 


0.5327 


0.5259 


0.5192 


0.5126 


0.5061 


0.4997 


0.4934 


172 


9 


0.5284 


0.5217 


0.5151 


0.5085 


0.5021 


0.4957 


0.4894 


171 


10° 


-0.5237 


-0.5171 


-0.5105 


— 0.5040 


-0.4977 


-0.4914 


-0.4851 


170° 


II 


0.5186 


0.5120 


0.5055 


0.4991 


0.4928 


0.4865 


0.4803 


169 


12 


0.5131 


0.5065 


0.5000 


0.4937 


0.4874 


0.4813 


0.4752 


168 


13 


0.5069 


0.5005 


0.4940 


0.4878 


0.4816 


0.4756 


0.4694 


167 


14 


0.5004 


0.4940 


0.4877 


0.4815 


0.47S4 


0.4695 


0.4635 


166 


15 


— 0.4934 


-0.4872 


-0.4805 


-0.4749 


-0.4688 


— 0.4630 


-0.4569 


165 


16 


0.4861 


0.4799 


0.4738 


0.4678 


0.4618 


0.4560 


0.4503 


164 


17 


0.4781 


0.4720 


0.4662 


0.4601 


0.4543 


0.4437 


0.4430 


163 


18 


0.4699 


0.4640 


0.4581 


0.4522 


0.4465 


0.4410 


0.4354 


162 


19 


0.4613 


0.4554 


0.4497 


0.4442 


0.4383 


0.4329 


0.4274 


161 



OCEAN TIDES 



21 



Table III {Continued) 
SOLAR TIDAL WAVE 



3 


a' 23200 


a' 23300 


a' 23400 


a' 23500 


a' 23600 


a' 23700 


a' 23800 


ft 


20° 


-0.4524 


— 0.4466 


-0.4409 


-0.4358 


— 0.4208 


-0.4244 


— 0.4190 


160° 


21 


0.4434 


0.4371 


0.4316 


0.4268 


0.4207 


0.4155 


0.4102 


159 


22 


0.4340 


0.4275 


0.4221 


0.4172 


0.4114 


0.4063 


0.401 1 


158 


23 


0.4242 


0.4174 


0.41 22 


0.4073 


0.4017 


0.3968 


0.3917 


157 


24 


0.4136 


0.4072 


0.4020 


0.3969 


0.3919 


0.3869 


0.3821 


156 


25 


-0.4023 


-0.3965 


-0.3913 


— 0.3864 


-0.3814 


-0.3767 


-0.3720 


155 


26 


0.3909 


0.3855 


0.3804 


0.3756 


0.3709 


0.3662 


0.3616 


154 


27 


0.3791 


0.3742 


0.3692 


0.3648 


0.3599 


0.3555 


0.3510 


153 


28 


0.3673 


0.3626 


0.3579 


0.3534 


0.3489 


0.3445 


0.3402 


152 


29 


0.3551 


0.3506 


0.3460 


0.3419 


0.3374 


0.3332 


0.3290 


151 


30 


-0.3427 


-0.3385 


-0.3341 


-0.3299 


-0.3257 


-0.3216 


-0.3176 


150 


31 


0.3302 


0.3260 


0.3218 


0.3180 


0.3138 


0.3099 


0.3060 


149 


32 


0.3175 


0.3135 


0.3095 


0.3056 


0.3017 


0.2979 


0.2941 


148 


33 


0.3045 


0.3006 


0.2967 


0.2930 


0.2892 


0.2857 


0.2820 


147 


34 


0.2913 


0.2875 


0.2838 


0.2802 


0.2767 


0.2730 


0.2698 


146 


35 


-0.2779 


-0.2743 


-0.2707 


-0.2673 


— 0.2639 


— 0.2607 


-0.2574 


145 


36 


0.2643 


0.2609 


0.2576 


0.2543 


0.2511 


0.2479 


0.2448 


144 


2>7 


0.2505 


0.2471 


0.2441 


0.2410 


0.2379 


0.2349 


0.2320 


143 


38 


0.2367 


0.2336 


0.2305 


0.2277 


0.2249 


0.2220 


0.2192 


142 


39 


0.2236 


0.2196 


0.2170 


0.2144 


0.2115 


0.2088 


0.2062 


141 


40 


— 0.2086 


— 0.2060 


— 0.2033 


— 0.2008 


— 0.1982 


-0.1957 


-0.1932 


140 


41 


0.1943 


0.1919 


0.1894 


0.1970 


0.1847 


0.1823 


0.1800 


139 


42 


0.1802 


0.1778 


0.1755 


0.1733 


0.1711 


0.1689 


0.1667 


138 


43 


0.1659 


0.1637 


0.1616 


0.1596 


0.1576 


0.1555 


0.1535 


137 


44 


0.1515 


0.1496 


0.1477 


0.1458 


0.1439 


0.1421 


0.1403 


136 


45 


-0.1371 


-0.1354 


-0.1337 


— 0.1320 


-0.1303 


— 0.1286 


— 0.1269 


135 


46 


0.1227 


0.1211 


0.1196 


0.1181 


0.1166 


0.1151 


0.1134 


134 


47 


0.1084 


0.1070 


0.1057 


0.1044 


0.1031 


0.1017 


O.IOOI 


133 


48 


0.0941 


0.0929 


0.0918 


0.0906 


0.0894 


0.0883 


0.0872 


132 


49 


0.0800 


0.0787 


0.0779 


0.0769 


0.0759 


0.0750 


0.0740 


131 


50 


—0.0657 


— 0.0648 


— 0.0640 


— 0.0632 


— 0.0624 


— 0.0616 


— 0.0608 


130 


51 


0.0516 


0.0508 


0.0503 


0.0497 


0.0489 


0.0485 


0.0478 


129 


52 


0.0376 


0.0371 


0.0367 


0.0362 


0.0357 


0.0353 


0.0348 


128 


55 


0.0240 


0.0233 


0.0231 


0.0227 


0.0225 


0.0223 


0.0220 


127 


5^ 


— 0.0104 


— 0.0097 


— 0.0096 


-0.0093 


— 0.0095 


— 0.0094 


— 0.0093 


126 


55 


+0.0032 


+0.0036 


+0.0037 


+0.0038 


+0.0034 


+0.0033 


+0.0032 


125 


56 


0.0163 


0.0168 


0.0165 


0.0168 


0.0161 


0.0159 


0.0157 


124 


57 


0.0297 


0.0298 


0.0294 


0.0295 


0.0287 


0.0283 


0.0281 


123 


58 


0.0430 


0.0427 


0.0421 


0.0418 


0.041 1 


0.0406 


0.0401 


122 


59 


0.0560 


0.0552 


0.0545 


0.0540 [ 


00532 


0.0525 


0.0520 


121 1 



22 



OCEAN TIDES 



Table III {Continued) 
SOLAR TIDAL WAVE 



;^ 


a' 23200 


a' 2330J 


a' 23403 


a' 23500 


a' 23600 


a' 23700 


a' 23800 


,^ 


00° 


+ 0.0686 


+0.0677 


+ 0.0668 


+0.0660 


+0.0651 


+0.0643 


+0.0635 


120° 


61 


0.0809 


0.0798 


0.0788 


0.0778 


0.0768 


0.0759 


0.0749 


119 


62 


0.0930 


0.0918 


0.0906 


0.0895 


0.0883 


0.0873 


0.0861 


118 


63 


0.1047 


0.1033 


0.1020 


0.1009 


0.0994 


0.0982 


0.0969 


117 


64 


0.1162 


0.1147 


0.1132 


0.1118 


0.1 104 


0.1090 


0.1076 


116 


65 


+ 0.1273 


+0.1256 


+ 0.1241 


+ 0.1225 


+0.1210 


+0.1195 


+0.1178 


115 


66 


0.1382 


0.1363 


0.1347 


0.1330 


0.1313 


0.1297 


0.1279 


114 


67 


0.1486 


0.1466 


0.1449 


0.1430 


0.1412 


0.1395 


0.1375 


113 


68 


0.1588 


0.1568 


0.1548 


0.1528 


0.1509 


0.1490 


0.1471 


112 


69 


0.1686 


0.1665 


0.1644 


0.1621 


0.1602 


0.1582 


0.1562 


III 


70 


+ 0.1781 


+0.1758 


+0.1736 


+0.1713 


+0.1692 


+0.1671 


+0.1650 


110 


71 


0.1871 


0.1847 


0.1824 


0.1800 


0.1777 


0.1755 


0.1733 


109 


72 


0.1957 


0.1932 


0.1908 


0.1883 


0.1859 


0.1836 


0.1813 


108 


73 


0.2039 


0.2014 


0.1989 


0.1962 


0.1938 


0.1912 


0.1889 


107 


74 


0.2117 


0.2091 


0.2065 


0.2037 


0.2012 


0.1986 


0.1962 


106 


75 


+ 0.2191 


+ 0.2164 


+0.2137 


+0.2108 


+0.2082 


+0.2054 


+0.2030 


105 


76 


0.2262 


0.2233 


0.2204 


0.2176 


0.2148 


0.2121 


0.2095 


104 


77 


0.2327 


0.2297 


0.2268 


0.2238 


0.2210 


0.2182 


0.2155 


103 


78 


0.2388 


0.2357 


0.2328 


0.2297 


0.2268 


0.2240 


0.2212 


102 


79 


0.2444 


0.2412 


0.2382 


0.2351 


0.2321 


0.2292 


0.2263 


101 


80 


+ 0.2495 


+0.2463 


+ 0.2432 


+0.2401 


+0.2370 


+0.2341 


+0.2311 


100 


81 


0.2541 


0.2509 


0.2478 


0.2445 


0.2414 


0.2384 


0.2354 


99 


82 


0.2583 


0.2551 


0.2519 


0.2486 


0.2454 


0.2424 


0.2393 


98 


83 


0.2620 


0.2587 


0.2555 


0.2520 


0.2489 


0.2458 


0.2427 


97 


84 


0.2653 


0.2619 


0.2586 


0-2553 


0.2521 


0.2489 


0.2458 


96 


85 


+ 0.2681 


+0.2646 


+ 0.2613 


+0.2579 


+0.2547 


+0.2514 


+0.2482 


95 


86 


0.2703 


0.2669 


0.2635 


0.2601 


0.2568 


0.2536 


0.2504 


94 


87 


0.2721 


0.2686 


0.2652 


0.2617 


0.2585 


0.2552 


0.2519 


93 


88 


0.2733 


0.2698 


0.2664 


0.2630 


0.2597 


0.2564 


0.2532 


92 


89 


0.2740 


0.2706 


0.2671 


0.2636 


0.2604 


0.2571 


0.2539 


91 


90 


+0.2743 


+0.2708 


+ 0.2674 


+ 0.2640 


+0.2606 


+0.2573 


+ 2.0541 


90 



/ 



D LI 

OCEAN .--=ife)ES 23 

The whole of the mathematical Itheory of the tides is illustrated 
in the two tables given above; and it is now only necessary to explain 
their properties and show how to use them. If we examine the diagram 
we shall perceive that the argument ^ of the tables denotes the angular 
distance at the center of the Earth between the disturbing body and the 
attracted point P. In other words the angle jS denotes the zenith 
distance of the attracting body from the attracted point. If we sup- 
pose the point F to be at any seaport town or harbor, and compute 
the zenith distances of the Sun and Moon at any time we shall have 
the argument ^ of the tables for that time and place; and by entering 
the tables with the arguments thus found, we can take out the heights 
of the two tidal waves by mere inspection. 

I will now give a practical illustration of the use of the tide tables, 
by computing the purely theoretical tides for a given port on the 
Earth. The islands of Great Britain and Ireland are perhaps more 
favorably situated for a complicated system of tidal waves than any 
other portion of the globe; and for that reason I have selected the 
port of London, England, for that purpose. 

If we assume the latitude of London to be +51° 30', the data 
in the Nautical Almanac for the year 191 9, will give for the first three 
days in October, at intervals of three hours, the values of (3, or the 
zenith distances of the Moon and Sun at London, as in the second 
and third columns of Table IV. At that time the Moon^s distance 
from the Earth is about 62.2. If we now enter the lunar tidal table, 
the columns headed a' = 62.0 and a^ = 62.5, with the argument (3 
for the Moon in Table IV, we can easily take out the height of the 
lunar tidal wave as in the fourth column of Table IV. 

Since the Sun on October i, is very nearly at its mean distance 
from the Earth we may assume a' = 23500 in the table for solar tides; 
and then with the argument /3 in Table IV we may easily take out the 
height of the solar tidal wave as in the fifth column of Table IV; 
and the -sum of the lunar and solar tidal waves gives the whole tide 
as in the last column of Table IV. According to Table IV, the highest 
tides at London during the first three days of October will take place 
on October 3, at three o'clock P. M., and would rise to a height of 
only 0.685 f^^t or 8.22 inches, unless modified by local conditions at 
that port. It would then be followed by the lowest ebb of 0.395 f^^t 
or 4.74 inches below the normal surface of the ocean; and the difference 
between high water and low water on that day would amount to only 
about thirteen inches. 

The construction of Table IV illustrates the ease and facility with 
which tidal tables for any port may be constructed for the free and 
undisturbed action of the tidal forces; and when the tidal constants 
for such ports have been determined it will be possible to tabulate in 
advance, the true effect of the tidal forces at any period of time. The 



24 



\ 



OC 



N 



Vi 



nt tides 



problem of the Tides, therefore, instead of being the most difficult, 
becomes by reason of these tables one of the easiest in the domain of 
practical astronomy. 



Table IV 
Tides at London, England, in October, 1919. 



Date 
Oct. 1919 






Moon tide 


Sun tide 


Whole tide 


Day Hour 


I O 


99° 13' 


54° 27' 


+ 0.492 


— 0.005 


+0.487 


3 


78 19 


67 41 


+0.473 


+0.139 


+0.612 


6 


72 24 


93 56 


+0.392 


+0.263 


+0.655 


9 


85 5 


120 


+0.527 


+0.066 


+0.593 


12 


108 36 


131 30 


+0.367 


— 0.084 


+0.283 


15 


135 20 


117 30 


-0.281 


+0.095 


-0.186 


i8 


148 21 


90 54 


-0.623 


+0.264 


-0.359 


21 


132 48 


65 30 


-0.213 


+0.128 


— 0.085 


2 O 


105 26 


54 48 


+0.413 


+0.003 


+0.416 


3 


81 42 


68 3 


+0.508 


+0.153 


+0.661 


6 


70 29 


94 16 


+0.367 


+0.263 


+0.630 


9 


78 39 


120 22 


+0.481 


+0.062 


+0.543 


12 


100 44 


131 56 


+0.473 


— 0.090 


+0.383 


15 


127 3 


117 52 


-0.055 


+0.090 


+0.035 


i8 


145 44 


91 9 


-0.558 


+0.264 


-0.294 


21 


136 38 


65 48 


— 0.320 


+0.131 


— 0.189 


3 o 


no 57 


55 12 


+0.318 


+0.006 


+0.324 


3 


85 20 


68 26 


+0.528 


+0.157 


+0.685 


6 


69 28 


94 37 


+0.352 


+0.261 


+0.613 


9 


72 17 


120 45 


+0.398 


+0.057 


+0.455 


12 


135 58 


132 16 


-0.300 


-0.095 


-0.395 


15 


118 7 


118 2 


+0.170 


+0.090 


+ 0.260 


i8 


140 7 


91 20 


— 0.408 


+0.263 


— 0.145 


21 


139 2 


66 4 


+ 0.382 


+0.133 


+0.515 


4 o 


117 30 


55 36 


+0.181 


+0.012 


+ 0.193 



GENERAL CONSIDERATIONS 

Newton's solution of the problem of the Tides given in the Prin- 
cipia is entirely erroneous; for there is no analogy between the action 
of bodies from the outside, and those which are inherent in the body 
itself. But his solution given in his Systema Mundi is correct in prin- 
ciple, but mathematically defective, inasmuch as it fails to give the 
direction in which the disturbing force acts. He there computes that 
the water which is directly under the Su7i would be 9 1-3 inches higher 
than the water 90° distant from that point; whereas, according to 
Table III, it should be that much lower. 

Again, Newton in the Principia computes the Moon's diameter 
which is directed towards the Earth to be 186 feet longer than the 
diameter perpendicular to it; but an exact computation shows that 



LONDON (London Bridge), ¥GLAND, 1919. 
Predicted Time and Height op Ah and Low Water. 



803 







JULY. 






AUGUST. '^^^ 


r-\ 


SEPTEMBER. 


Day 


Mish 


Low Hi£5h 


Low 


Day 


High 


Low 


High 


Lov^' 


Bay 


Low 


High 


Low 


Hit:h 


1 


3:57 


10:50 4:li 


ii:i5 


1 


o:Oti 


11:4? 


5:18 




■J. 


0:08 


5:43 


12:20 


6:01 


Tu 


21.1 


0.8 2i.G 


-0.5 


1' 


20.2 


2.0 


i;O.S 




M 


2.4 


VI 1 


S.5 


19,4 


2 


4:44 


11:35 5:00 


11:58 




Low 


High 


Low 


High 


2 


0:41 


6:27 


12:55 


6:49 


W 


20.5 


1.2 20.9 


0.0 


2 


0:09 


5:42 


12:24 


5:56 


Tu 


3,5 


18.2 


4.4 


18.3 


3 


5:32 


12:18 5:47 




Sa 


1.0 


19.3 


2.9 


19.8 


3 


1:21 


7:18 


1:42 


7:46 


Th 


19.6 


2.1 20.0 


. . . 


3 


0:46 


6:26 


1:01 


6:41 


W 


4,6 


17.3 


6.2 


17.3 




Low 


High Low 


High 


Su 


2.1 


18.4 


8.9 


18.8 


4 


2:12 


8:17 


2:46 


8:50 


1 * 


0.42 


6:21 1:01 


6:35 


4 


1:25 


7:13 


1:42 


7:80 


Th 


5.5 


16.7 


6.7 


16.7 


F 


0.9 


18.7 3.1 


19.0 


M 


3.3 


17.5 


4.8 


17.9 


5 


3:21 


9:20 


4:01 


9:55 


5 


1:25 


7:10 1:46 


7:25 


5 


2:09 


8:05 


2:31 


8:25 


F 


6,2 


16.5 


5.7 


16.6 


Sa 


2.1 


17. 7 4.2 


18.1 


Tu 


4.4 


16.8 


5.6 


17.1 


6 


4:32 


10:23 


5:08 


10:57 


6 


2:11 


8:00 2:34 


8:15 


6 


3:02 


8:59 


3:31 


9:24 


Sa 


6.2 


18.8 


5.1 


17.1 


Su 


3.2 


17.0 5.1 


17.3 


W 


5.2 


16.5 


6.9 


16.7 


7 


5:34 


* 11:22 


6:06 


11:53 


' 


3:01 


8:51 3:26 


9:08 


7 


4:03 


9:57 


4:36 


10:25 


Su 


5.6 


17.7 


4.1 


18,1 


M 


4.1 


16.5 5.6 


16.9 


Th 


5.6 


16.5 


5.6 


16.7 


8 


6-29 


12:15 


6 '58 




8 


3:53 


9:43 4.21 


10:03 


8 


5:04 


10:56 


5:37 


11:24 


M 


4,8 


19.0 


3.0 




Tu 


4.7 


16.4 5.7 


16.8 


F 


5.5 


17.0 


5.0 


17.2 




High 

0:44 


Low 


High 
1:03 


Low 


9 


4:47 


10:36 5:15 


10:57 


9 


6:02 


11:51 


6:34 




9 


7:19 


7:46 


W 


4.8 


16.7 5.3 


17.0 


Sa 


5.1 


17.9 


4.0 




Tu 


19.3 


4.0 


20.3 


2.1 


10 


5:39 


11:28 6:07 


11:52 




High 


Low 


High 


Low 


10 


1:30 


8:05 


1:47 


8:83 


Th 


4.7 


17.4 4.6 


17.6 


10 


0:19 


6:56 


12:42 


7:26 


w 


20.4 


3,1 


21.4 


1.3 


11 


6:30 


12:19 6:59 




Su 


18.1 


4.4 


19.0 


3.0 


11 


2:10 


8:50 


2:29 


9:16 


F 


4.3 


18.2 8.8 




11 


1:09 


7:46 


1:29 


8:16 


Th 


21.3 


2.4 


22.3 


0.7 




High 


Low High 


Low 


M 


19.2 


3.8 


20.2 


2.1 


12 


2:54 


9:. 33 


3.10 


9:.59 


12 


0:43 


7:20 1:06 


7:50 


12 


1:54 


8:34 


2:12 


9:02 


F 


21.9 


1.8 


22.8 


0.4 


Sa 


18.4 


3.9 19.2 


3.0 


Tu 


20.1 


8.1 


21.1 


l.S 


18 


3:34 


10:15 


3:51 


10:41 


13 


1:30 


8:11 1:51 


8:40 


IS 


2:37 


9:19 


2:53 


9:46 


Sa 


22.1 


1.5 


22.7 


0.5 


Su 


19.3 


3.5 20.1 


2,2 


w 


20.8 


2.6 


21.8 


0.7 


14 


4:16 


10:56 


4:34 


11:22 


14 


2:13 


8:59 2:33 


8:27 


14 


3:18 


10:01 


3:34 


10:28 


Su 


21.8 


1.7 


22.1 


1.1 


M 


19.9 


3. 1 20.8 


1.5 


Th 


21.3 


2.2 


22.2 


0,5 


15 


5:00 


11:40 


5:22 




15 


2:59 


9:44 3:14 


10:11 


15 


3:59 


10:39 


4:14 


11:04 


M 


21.0 


2.2 


21.0 




Tu 


20.4 


2.8 21.2 


1.0 


F 


21.3 


2.1 


22.0 


0.6 




Low 


High 

5:51 


Low 


High 
6:17 


16 


3:41 


10:25 3:55 


10:51 


16 


4:41 


11:17 


4:55 


11:43 


16 


0:08 


12:28 


W 


20.5 


2.6 21.3 


0.8 


Sa 


20.9 


2.3 


21.5 


1.1 


Tu 


2.0 


19,8 


3.0 


19.5 


17 


4:23 


11:02 4:36 


11:29 


17 


5:25 


11:55 


5:41 




17 


1:00 


6:48 


1:27 


7:22 


Th 


20.3 


2.7 21.0 


1.0 


Su 


20.2 


2.8 


20.5 


. . , 


W 


3.3 


18.6 


4.1 


18,1 


18 


5:06 


11:39 5:19 






Low 


High 


Low 


High 


18 


2:05 


7:55 


2:40 


8:33 


F 


19.9 


3. 20.5 


. 


18 


0:24 


6:14 


12:39 


6:34 


Th 


4.6 


17.5 


4.9 


17,2 




Low 


High Low 


High 


M 


2.0 


19.3 


3.6 


19,3 


19 


3:19 


9:05 


3:56 


9:44 


19 


0:06 


5:51 12:16 


6:05 


19 


1:12 


7:11 


1:35 


7:38 


F 


5.3 


17.1 


4.9 


16.9 


Sa 


1.5 


19.2 3.5 


19.7 


Tu 


3.2 


18.3 


4.5 


18,1 


20 


4:32 


10:13 


5:06 


W:r,2 


20 


0:46 


6:42 12:59 


6:58 


20 


2:16 


8:15 


2:50 


8:47 


Sa 


5.2 


17,4 


4,8 


17.4 


Su 


2.2 


18.5 4.2 


18.8 


W 


4.2 


17.4 


5.2 


17.2 


21 


5:37 


11:16 


6:06 


11:52 


21 


1:33 


7:39 1:52 


8:00 


21 


3:32 


9:24 


4:10 


9:58 


Su 


4.5 


18,0 


3.3 


18.3 


M 


3.0 


17.8 4.8 


18.0 


Th 


5.0 


17.1 


5.2 


17.0 


22 


6:33 


12:11 


6:58 




22 


2:33 


8:41 3:01 


9:06 


22 


4:46 


10:32 


5.22 


11:06 


M 


3.7 


19.0 


2.3 


. 


Tu 


3.9 


17.4 5.3 


17.5 


F 


4.8 


17.4 


4.5 


17.5 




High 


Low 


High 


Low 


23 


3:45 


9:45 4:22 


10:14 


23 


5:52 


11:34 


6:24 




23 


0:43 


7:23 


12:57 


7:46 


W 


4.3 


17. 4 5.1 


17.4 


Sa 


4.2 


18.2 


3.3 




Tu 


19.3 


3.0 


20.0 


1.6 


24 


4:58 


10:49 5:43 


11:20 




High 


Low 


High 


Low 


24 


1:25 


8:07 


1:89 


8:30 


Th 


4.2 


17.8 4.4 


17.9 


24 


0:08 


6:51 


12:30 


7:20 


W 


20.1 


2,4 


20.9 


1.2 


25 


6:04 


11:51 6:38 




Su 


18.4 


3.3 


19.4 


2.1 


25 


2:04 


8:50 


2:18 


9:11 


F 


3.6 


18.7 3.8 


. . . 


25 


1:01 


7:44 


1:20 


8:11 


Th 


20.7 


2,1 


21.6 


1.0 




High 


Low High 


Low 


M 


19.5 


2.5 


20.5 


1.2 


26 


2:40 


9:29 


2:54 


9:49 


26 


0:21 


7:05 12:46 


7:36 


26 


1:48 


8:33 


2:03 


8:58 


F 


21.1 


1,9 


21.9 


1.1 


Sa 


18.8 


2.8 19.8 


2.1 


Tu 


20.4 


1.8 


21.4 


0.5 


27 


3:17 


10:06 


3:31 


10:26 


27 


1:16 


8:01 1:38 


8:31 


27 


2:30 


9:18 


2:44 


9:42 


Sa 


21.2 


1.9 


21.8 


1.4 


Su 


19.8 


2. 1 20.9 


1.0 


W 


21.0 


1.5 


22.0 


0.2 


28 


3:53 


10:40 


4:08 


11:01 


28 


2:06 


8:54 2:24 


9:21 


28 


3:09 


9:59 


3:23 


10:22 


Su 


21.0 


2.1 


21.5 


2.0 


M 


20.6 


1.5 21.6 


0.1 


Th 


21.3 


1.4 


22.1 


0.2 


29 


4:28 


11:14 


4:45 


11:84 


1^ 


2:51 


9:43 3:08 


10:08 


29 


3:47 


10:38 


4:01 


10:58 


M 


20,6 


2.5 


20,8 


2.8 


21.1 


1.0 22.0 


-0.4 


F 


21.1 


1.5 


21.9 


0.6 


30 


5:06 


11:47 


5:28 




30 


3:35 


10:27 3:50 


10:50 


30 


4:25 


11:14 


4:39 


11:34 


Tu 


19.8 


3.1 


19.8 


.* '. .. 


W 


21.2 


1.0 22.0 


-0.4 


Sa 


20.7 


2.0 


21.3 


1.4 












31 


4:18 


11:08 4:32 


11:30 


31 


5:03 


11:47 


5:18 














Th 


20.9 


1.3 21.5 


0.1 


Su 


20.0 


2.7 


20.5 














Tim 


e meridian, 0°. Heavy-faced 


I tyi^e 


indicates p. ra 


tides. 


Heigl 


its are 


reckoned from 


approximate 


mean] 


ow water springs, the 


daturr 


. oi' sou 


ndings on Admiralty Charts 


See 


p. 7, 









304 






LONDON (Lod/ /Bridge 


), ENGLAND, 


1919. 


1 






Predicted Time an] 


f 7IGHT OF High and Low Water. 


OCTOBEK. 


/ 


j„ ioVEMBER. 


1 DECEMBER. 


Day 


Lo-vv 


High 


Low High 


Day 


Low 


High 


Low High 


Day 


Low 


High Low Hiah 


1 


0:06 


5:48 


12:24 6:15 


1 


1:04 


7:03 


1:40 7:48 


1 


1:26 


7:30 2:05 8:15 


"\V 


3.7 


18.9 


3.8 18.7 


Sa 


5.6 


17.7 


4.5 17.2 


M 


5.8 


17.9 4.2 17.3 


2 


0:43 


6:38 


1:08 7:12 


2 


2:04 


8:08 


2:46 8:52 


2 


2:25 


8:33 3:08 9;15 


Th 


4.7 


17.9 


4.6 17.6 


Su 


6.3 


17.2 


4.9 16.9 


Tu 


6.2 


17.6 4.5 17.3 


3 


1:33 


7:38 


2:10 8:19 


3 


3:15 


9:12 


3:54 9:53 


3 


3:34 


9:35 4:12 10.14 


F 


5.7 


17.1 


5.3 16.8 


M 


6.6 


17.2 


4.8 17.2 


Y/ 


6.2 


17.7 4.4 17.7 


4 


2:42 


8:44 


3:25 9:24 


4 


4:23 


10:13 


4:55 10:51 


4 


4:40 


10:36 5:12 lltll 


Sa 


6.5 


16.7 


5.5 16.7 


Tu 


6.2 


17.7 


4.2 18.0 


Th 


5.6 


18.1 3.9 18.5 


5 


3:58 


9:49 


4:35 10:27 


5 


5:21 


11:11 


5:49 11:44 


5 


5:41 


11:34 6:09 . . . 


Su 


6.6 


16.9 


6.0 17.1 


W 


5.4 


18.5 


3.4 19.1 


F 


4.7 


19.0 3.3 .. . 


6 


5:03 


10:50 


6:34 11:24 


6 


6:13 


12:05 


6:40 . . . 




High 


Low High Low 


M 


6.0 


17.7 


4.1 18.0 


Th 


4.4 


19.7 


2.6 .. . 


6 


0:06 


6:39 12:30 7:04 


7 


5:58 


11:44" 


6:25 . . . 




High 


Low 


High Low 


Sa 


19.6 


3.7 20.0 2.6 


Tu 


5.1 


18.8 


3.1 .. . 


7 


0:33 


7:02 


12:54 7:30 


7 


0:58 


7:36 1:21 8:01 




High 


Low 


High Low 


F 


20.2 


3.4 


20.9 1.9 


Su 


20.8 


2.6 21.0 2.0 


8 


0:15 


6:48 


12:35 7:14 


8 


1:21 


7:53 


1:42 8:20 


8 


1:46 


8:32 2:11 8:56 


W 


19.3 


4.1 


20.1 2.2 


Sa 


21.4 


2.4 


21.9 1.4 


M 


21.8 


1.6 21.7 1.4 


9 


1:02 


7:34 


1:20 8:00 


9 


2:06 


8:47 


2:28 9:12 


9 


2:34 


9:26 3.:00 9:51 


Th 


20.5 


3.1 


21.4 1.4 


Su 


22.2 


1.6 


22.5 1.0 


Tu 


22.5 


0.7 22.1 1:0 


10 


1:47 


8:21 


2:04 8:47 


10 


2:51 


9:38 


3:14 10:03 


10 


3:22 


10:18 3:49 10:42 


F 


21.6 


2.3 


22.4 0.9 


M 


22.7 


1.0 


22.6 0.9 


w 


22.7 


0.1 22.0 1.0 


11 


2:28 


9:07 


2:48 9:34 


11 


3:37 


10:30 


4:03 10:54 


11 


4:10 


11:07 4:40 11:31 


Sa 


22.3 


1.6 


22.9 0.6 


Tu 


22.6 


0.7 


22.1 1.2 


Th 


22.4 


-0.1 21.4 1.3 


12 


3:11 


9:54 


3:31 10:20 


12 


4:25 


11:20 


4:54 11:45 


12 


4:59 


11:57 6:32 . . . 


Su 


22.6 


1.2 


22.9 0.7 


W 


22.0 


0.8 


21.2 1.8 


F 


21.7 


0.2 20.4 . . . 


13 


3:55 


10:41 


4:17 11:07 


13 


5:17 


12:11 


5:50 . . . 




Low 


High Low High 
5:51 12:45 6:28 


M 


22.4 


1.2 


22.3 1.2 


Th 


21.0 


1.3 


20.0 . . . 


13 


0:20 


14 


4:41 


11:30 


6:07 11:56 




Low 


High 


Low High 


Sa 


2.1 


20.6 1.0 19,4 


Tu 


21.6 


1.6 


21.1 2.1 


14 


0:39 


6:13 


1:05 6:52 


14 


1:10 


6:45 1:36 7:24 


15 


5:32 


12:22 


6:03 . . . 


F 


2.8 


19.8 


2.2 18.8 


Su 


3.2 


19.4 2.1 18.3 


W 


20.4 


2.4 


19.6 . . . 


15 


1:35 


7:14 


2:04 7:55 


15 


2:02 


7:41 2:31 8:21 




Low 


High 

6:31 


Low High 
1:20 7:08 


Sa 


3.8 


18.7 


3.0 17.9 


M 


4.3 


18.4 3.1 17.5 


16 


0:51 


16 


2:36 


8:16 


3:06 8:58 


16 


2:59 


8:37 8:28 9:16 


Th 


3.3 


19.1 


3.3 18.3 


Su 


4.8 


17.9 


3.7 17.4 


Tu 


6.2 


17.6 4.0 17.0 


17 


1:53 


7:36 


2:26 8:17 


17 


3:40 


9:17 


4:09 9:58 


17 


3:57 


9:32 4:23 10:10 


F 


4.4 


18.0 


4.1 17.5 


M 


5.3 


17.5 


3.9 17.4 


W 


5.7 


17.1 4.5 16.8 


18 


3:02 


8:43 


3:36 9:26 


18 


4:40 


10:14 


5:05 10:52 


18 


4:52 


10:25 6:14 11:01 


Sa 


5.2 


17.4 


4:3 17.2 


Tu 


5.3 


17.5 


3.9 17.5 


Th 


5.7 


17.1 4.6 17.0 


19 


4:11 


9:48 


4:42 10:30 


19 


5:33 


11:08 


5.56 11:41 


19 


5:43 


11:18 6:05 11:48 


Su 


5.2 


17.2 


4.0 17.4 


W 


5.0 


17.8 


3.7 18.0 


F 


5.4 


17.4 4.6 17.6 


20 


5:13 


10:49 


5:40 11:28 


20 


6:20 


11:55 


6:42 . . . 


20 


6:31 


12:07 6:52 . . . 


M 


4.8 


17.8 


3.4 18.0 


Th 


4.6 


18.3 


3.5 .. . 


Sa 


4.8 


18.0 4.3 .. . 


21 


6:08 


11:43 


6:30 . . . 




High 


Low 


High Low 




High 


Low High Low 


' Tu 


4.2 


18.5 


2.8 .. . 


21 


0:24 


7:04 


12:39 7:25 


21 


0:33 


7:17 12:5« 7:36 










F 


18.6 


4.1 


19.1 3.3 


Su 


18.5 


4.2 18.7 4.0 




High 


Low 


High Low 
















22 


0:15 


6:54 


12:29 7:16 


22 


1:03 


7:46 


1:21 8:06 


22 


1:16 


8:02 1:37 8:21 


W 


18.7 


3.7 


19.4 2.4 


Sa 


19.4 


3.7 


19.8 3.2 


M 


19.4 


3.4 19.6 3.7 


23 


0:57 


7:37 


1:10 7:58 


23 


1:43 


8:28 


2:01 8:47 


23 


1:57 


8:47 2:20 9:05 


Th 


19.5 


3.3 


20.1 2.2 


Su 


20.1 


3.1 


20.4 3.0 


Tu 


20.3 


2.7 20.2 3.3 


24 


1:34 


8:18 


1:49 8:39 


24 


2:21 


9:10 


2:40 9:28 


24 


2:38 


9:31 8:01 9:48 


F 


20.2 


2.8 


20.8 2.1 


M 


20.7 


2.6 


20.8 3.0 


W 


21.0 


2.0 20.6 8.0 


25 


2:11 


8*57 


2:27 9*18 


25 


2:59 


9:50 


3:20 10:08 


25 


3:18 


10:13 3:44 10:28 


Sa 


20.8 


2.5 


21.2 2.1 


Tu 


21.1 


2.2 


20.9 3.0 


Th 


21.3 


1.5 20.7 3.0 


26 


2:47 


9:35 


3:04. 9:54 


26 


3:38 


10:29 


4:02 10:45 


26 


3:58 


10:53 4:26 11:05 


Su 


2i.i 


2.4 


2i.5 2.3 


W 


21.1 


2.0 


20.7 3.2 


F 


21.3 


1.3 20.4 3.2 


27 


3:23 


10:12 


3*42 10-30 


27 


4:17 


11:08 


4:44 11:23 


27 


4:38 


11:30 5:09 11:40 


M 


21.1 


2.2 


21.3 2.6 


Th 


20.8 


2.1 


20.1 3.7 


Sa 


21.0 


1.5 19.9 3.6 


28 


4:00 


10 '48 


4:21 11:05 

20.7 3.1 


28 


4:58 


11:47 


6:30 . . . 


28 


5:20 


12:07 5:53 . . . 


Tu 


20.9 


2.3 


F 


20.2 


2.4 


19.3 . . . 


Su 


20.4 


1.9 19.1 . . . 


29 


4:39 
20.4 


11*24 


5:02 11:40 
20.0 3.8 




Low 


High 


Low High 




Low 


High Low High 


W 


2.6 


29 


0:00 


5:43 


12:28 6:20 


29 


0:15 


6:04 12:45 6:42 


80 

Th 


Sa 


4.3 


19.4 


2.9 18.4 


M 


4.2 


19.5 2.7 18.4 


5:20 
19.6 


12:02 
3.2 


5:51 . . . 
18.9 . . . 


30 


0:39 


6:32 


1:12 7:16 


30 


0:53 


6:55 1:28 7:37 




Su 


5.0 


18.6 


3.6 17.7 


Tu 


4.8 


18.7 3.4 17.7 


31 

F 


Low 

0:19 
4.7 


High 

6:06 
18.6 


Low High 

12:46 6:46 

3.9 17.9 










31 


1:40 
5.4 


7:53 2:20 8:37 
18.0 4.1 17.4 


Time 


raeridi 


in, 0°. 


IlGvT.vv-faced 


type i 


ndicate 


3 p. m. 


tides. Heigt 


ts are 


reckoned from approximate | 


mean Ic 


)w water springs, the datum 


of soui 


idings on Admiralty Charts. 


Seex 


). 7. 





OCEAN TIDES 25 

the Moon's diameter which is directed towards the Earth is 126.66 
feet shorter than the diameter at right angles to it. 

A surprising fatuity seems to have attended all of Newton's 
efforts to solve the problem of the tides; but the great celebrity he 
had acquired as a mathematician, placed him above the reach of pop- 
ular criticism; and his erroneous solution has been universally accepted 
as correct. And while we cheerfully admit that Newton made the 
most brilliant and important discovery of the principle and law of 
universal gravitation, we must also concede that, in his treatment 
of the tidal problem, he was the author of the most misleading fallacy 
that was ever imposed upon a trusting world in the name of science; 
so that even now, after the lapse of more than two hundred and thirty 
years, mathematicians and astronomers still hug the delusion as if its 
preservation and perpetuity were essentially important for the con- 
tinued existence and improvement of tidal science. 



Mo^ 



